What are Physics-Informed Neural Networks (PINNs): A Complete Guide to Scientific Machine Learning in 2025
- Muiz As-Siddeeqi
- 6 days ago
- 24 min read
Traditional machine learning often treats systems as black boxes, learning patterns without understanding the underlying rules. But what if you could teach a neural network the fundamental laws of physics before it ever sees a single data point? That's the revolutionary promise of Physics-Informed Neural Networks—a technology that's quietly reshaping how scientists and engineers solve problems once thought impossible.
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TL;DR
PINNs embed physical laws directly into neural network training, making them work with sparse data where traditional ML fails
First introduced in 2017 by Raissi, Perdikaris, and Karniadakis at Brown University
Solve partial differential equations (PDEs) without requiring mesh generation or massive datasets
Active in aerospace, healthcare, energy, and manufacturing with documented speedups of 100 million times over traditional methods
Key challenge: Computational cost and convergence issues for complex problems
Growing fast: Part of the $79 billion machine learning market projected to hit $500 billion by 2030
What Are Physics-Informed Neural Networks (PINNs)?
Physics-Informed Neural Networks (PINNs) are neural networks that incorporate known physical laws—described by differential equations—into their loss functions during training. This approach enables PINNs to solve complex physics problems with minimal data, providing accurate predictions while respecting fundamental scientific principles. They work by penalizing violations of physical laws alongside prediction errors.
Table of Contents
Understanding the Core Concept
Imagine you're trying to predict how heat flows through a complex metal engine part. Traditional machine learning would need thousands of temperature measurements at different locations and times. Traditional simulation methods would need you to divide the part into millions of tiny pieces and calculate temperatures at each one.
PINNs take a different path. They start by knowing the heat equation—the fundamental physics describing heat flow. During training, the neural network learns to predict temperatures while simultaneously respecting this physical law. If predictions violate the heat equation, the network gets penalized and adjusts itself.
The result is striking. PINNs can make accurate predictions with far fewer data points than conventional machine learning. They can also handle irregular geometries that make traditional simulation methods struggle.
At their core, PINNs are universal function approximators that embed physical knowledge into the learning process (Wikipedia, 2025). When you have partial differential equations (PDEs) describing a system, PINNs use these equations as constraints during training.
The key innovation: Instead of just minimizing the difference between predictions and measurements, PINNs minimize a combined loss that includes how well the prediction satisfies the governing physics equations.
The Birth of PINNs: A Brief History
The story of PINNs begins at Brown University in 2017. Three researchers—Maziar Raissi, Paris Perdikaris, and George E. Karniadakis—published two groundbreaking papers on arXiv in November 2017 that introduced the framework (arXiv:1711.10561 and arXiv:1711.10566).
November 2017: Raissi, Perdikaris, and Karniadakis release "Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations" and Part II on data-driven discovery.
2019: Their comprehensive paper "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations" appears in Journal of Computational Physics (Raissi et al., 2019). This paper has since been cited over 30,000 times.
The researchers built on earlier work from the 1990s. Lagaris and colleagues proposed using neural networks to solve differential equations in 1998, but the approach lacked the computational power and algorithmic sophistication that modern deep learning provides.
2021: PINNs enter the Gartner Hype Cycle for Emerging Technologies, signaling serious industry interest (Medium, 2024).
2022-2024: Explosive growth in PINN research. A Google Scholar survey shows rapid growth across scientific domains, with thousands of papers published annually (arXiv, 2025).
2025: NVIDIA rebrands its Modulus framework as PhysicsNeMo, offering commercial-grade tools for PINNs at scale.
The three original authors remain active. Maziar Raissi is Associate Professor at UC Riverside with over 31,000 citations. Paris Perdikaris is at University of Pennsylvania with over 43,000 citations. George Karniadakis is at Brown University leading the Division of Applied Mathematics.
How PINNs Actually Work
Let's break down the mechanics step by step.
Step 1: Define the Physics
You start with a differential equation describing your system. For example, the Navier-Stokes equations for fluid flow, the heat equation for thermal systems, or the Schrödinger equation for quantum mechanics.
Step 2: Set Up the Neural Network
A PINN uses a standard neural network architecture—typically a fully connected feedforward network with multiple layers. The network takes spatial coordinates (x, y, z) and time (t) as inputs and outputs the solution at those points.
Step 3: Create the Physics-Informed Loss
This is where PINNs differ from regular neural networks. The loss function has multiple components:
Data Loss: Measures how well predictions match any available measurements.
Physics Loss: Measures how well predictions satisfy the governing differential equation. This is computed using automatic differentiation to calculate derivatives of the network output.
Boundary/Initial Condition Loss: Measures violations of boundary and initial conditions.
Total Loss = α × Data Loss + β × Physics Loss + γ × Boundary Loss
The weights (α, β, γ) are hyperparameters that balance these terms.
Step 4: Train Using Backpropagation
Standard optimization algorithms (Adam, L-BFGS) adjust the network weights to minimize the total loss. Automatic differentiation in frameworks like PyTorch or TensorFlow calculates the needed derivatives.
Step 5: Query the Trained Network
Once trained, you can query the network at any point in the domain to get predictions. Unlike finite element methods, no mesh is required.
Key Advantage: The trained PINN is a continuous function. You can evaluate it anywhere in the domain and compute derivatives with respect to any parameter, making it ideal for inverse problems and optimization.
According to MATLAB (2025), PINNs were first introduced in 2017 and now have many variations including Bayesian PINNs for uncertainty quantification, Variational PINNs using weak forms, and First-Order formulated PINNs for higher-order PDEs.
Why PINNs Matter: The Data Scarcity Problem
Traditional deep learning has a dirty secret: it's data-hungry. Training a neural network to model complex phenomena typically requires thousands or millions of examples.
But in many scientific and engineering domains, data is expensive or impossible to get. Consider:
Medical imaging: Patient data is limited by privacy, ethics, and rarity of certain conditions
Climate modeling: We only have one Earth and limited historical records
Extreme environments: Nuclear reactors, deep ocean, outer space—measurements are costly
New materials: Testing novel compounds is time-intensive and expensive
This is where PINNs shine.
The Prior Knowledge Advantage: By embedding physical laws, PINNs constrain the space of possible solutions. Instead of searching through all mathematically possible functions, the network only considers functions that obey physics. This regularization dramatically reduces the data needed for accurate predictions (ScienceDirect, 2018).
A 2018 study in Journal of Computational Physics showed PINNs could accurately solve complex PDEs with just a handful of scattered measurements—scenarios where purely data-driven approaches fail completely.
Amplifying Information Content: The research team at Brown University explained it this way: "Encoding structured information into a learning algorithm results in amplifying the information content of the data that the algorithm sees, enabling it to quickly steer itself towards the right solution and generalize well even when only a few training examples are available" (ScienceDirect, 2018).
PINNs vs Traditional Methods
Advantages of PINNs
Meshfree Nature: Traditional numerical methods like finite element analysis require dividing the domain into a mesh. For complex 3D geometries, mesh generation can take weeks and requires expert knowledge. PINNs eliminate this bottleneck (Wikipedia, 2025).
Data Efficiency: Work with sparse, noisy, or incomplete data by leveraging physics as a regularizer.
Inverse Problem Capability: Easily estimate unknown parameters in differential equations from limited observations. The differentiability of neural networks makes optimization straightforward.
Continuous Solutions: Unlike grid-based methods, you can query the solution at any point without interpolation.
Forward and Inverse in One Framework: The same PINN setup solves both forward problems (predict behavior given parameters) and inverse problems (infer parameters from behavior).
Limitations of PINNs
Computational Cost: Training PINNs often requires many iterations—much longer than solving the same problem with established numerical methods. A single training run can take hours or days (Journal of Scientific Computing, 2022).
Accuracy Gap: Current PINNs typically don't achieve the same accuracy as high-order numerical methods for many problems.
Optimization Challenges: The complex loss landscape can lead to slow convergence or getting stuck in local minima. Balancing multiple loss terms is an art.
Limited to Known Physics: If you don't know the governing equations, standard PINNs won't help. However, hybrid approaches can discover equations from data.
Real-World Applications Across Industries
PINNs are no longer just academic curiosities. Industries are deploying them for real problems.
Aerospace and Aviation
Aircraft and spacecraft operate under extreme conditions where physical testing is prohibitively expensive. PINNs are being used for:
Airflow modeling: Zhang et al. (2024) applied PINNs to model turbulent airflow in enclosed spaces, improving prediction accuracy for pressure by 82.9%, horizontal velocity by 59.4%, and vertical velocity by 70.5% compared to classical methods.
Helicopter blade mechanics: Singh et al. (2024) demonstrated that PINNs significantly outperformed traditional artificial neural networks for modeling prismatic cantilever beams under loading—critical for helicopter blade design.
Mesh deformation: Aygun et al. (2023) used PINNs for adaptive mesh deformation with precise boundary condition adherence, important for computational fluid dynamics.
Cardiovascular Medicine
Heart disease is the leading cause of death globally. PINNs are transforming cardiovascular diagnostics:
Blood flow prediction: Kissas et al. (2020) introduced a PINN framework to model cardiovascular flows and predict arterial blood pressure using non-invasive MRI data. The neural networks captured fine details in propagating waveforms without invasive measurements.
Cardiac activation mapping: Sahli Costabal et al. (2020) developed PINNs for identifying underlying wave dynamics in cardiac tissue, helping diagnose arrhythmias.
Personalized heart modeling: Buoso et al. (2024) created parametric PINNs that simulate personalized left ventricular biomechanics from cardiac images, running computationally cheaper than finite element models while maintaining accuracy (arXiv, 2025).
Energy and Renewable Resources
The energy transition demands better predictive tools:
Wind power forecasting: A 2025 study in Scientific Reports presented a hybrid framework combining machine learning, MATLAB Simulink, and PINNs for wind power prediction in a 10kW permanent magnet synchronous generator system. The integrated approach achieved an R² of 0.998, significantly outperforming individual algorithms.
Geothermal systems: Feng et al. (2025) applied PINNs to thermal-hydraulic coupling in silty sands, demonstrating accurate simulation for geothermal engineering applications (International Journal of Geo-Engineering, 2025).
Manufacturing and Industrial Design
Air knife optimization: Kinetic Vision, a Cincinnati-based technology company, used NVIDIA PhysicsNeMo to optimize air knife designs for manufacturing. Engineer Michael Eidell coupled trained PINNs with SolidWorks CAD software, enabling real-time design feedback—changing trailing edge radius provided instant jet angle predictions (NVIDIA Technical Blog, 2022).
Chemical reactors: Shell achieved 100 million times faster inferencing for multiscale chemical reactors using physics-based surrogate models with PINNs powered by NVIDIA Modulus (NVIDIA Resources, 2024).
Electric motor design: Gholampour et al. (2024) investigated parameterized PINNs for natural convection problems in electric motors, though parameterizing three variables increased computational cost by 46%.
Materials Science
Composite laminates: A comprehensive December 2024 review in Mathematics journal highlighted PINNs' applications in laminated composites for structural health monitoring, stress-strain analysis, and failure prediction. PINNs addressed challenges like anisotropic behavior and multi-layered structures that trouble traditional methods (MDPI, 2024).
Environmental Science
Mosquito population dynamics: Researchers adapted PINNs to optimize ordinary differential equation (ODE) systems for mosquito population modeling—critical for disease control. The study addressed extreme multi-scale behavior in real-world ODE systems (PLOS ONE, 2024).
Documented Case Studies
Case Study 1: Hybrid Quantum PINNs for Fluid Flow
Organization: Sedykh et al. research team
Date: 2024
Problem: Modeling linear fluid flows in complex three-dimensional Y-shaped mixers
Approach: Developed Hybrid Quantum Physics-Informed Neural Networks (HQPINN) integrating classical and quantum computing
Results: Achieved 21% higher accuracy compared to classical neural networks for predicting velocity and pressure distributions
Source: Challenges and Issues of Modern Science, 2025
Case Study 2: SoftServe Virtual Metering for Oil Wells
Organization: SoftServe with oil industry client
Date: Presented at NVIDIA GTC 2024
Problem: Flow-metering in electrical submersible pump (ESP) oil wells is expensive and technically limited
Approach: Designed hybrid PINN-based virtual flow meter combining physics knowledge with auxiliary sensor data using NVIDIA Modulus
Results: Accurate flow rate estimates without costly physical equipment
Impact: Demonstrated practical industrial deployment of PINNs for real-time monitoring
Source: SoftServe Blog, March 2024
Case Study 3: Shell Chemical Reactor Simulation
Organization: Shell
Date: 2024
Problem: Multiscale chemical reactors require prohibitively expensive computational simulations
Approach: Used physics-based surrogate models with PINNs powered by NVIDIA Modulus
Results: Achieved 100 million times faster inferencing compared to traditional simulation methods
Impact: Enabled rapid design iteration and optimization previously impossible
Source: NVIDIA GTC 2024, Session S62060
Advanced PINN Variants
The field has exploded with specialized PINN variants addressing specific limitations:
Extended PINNs (XPINNs)
Purpose: Handle complex geometries by decomposing domains
How: Uses multiple neural networks in smaller subdomains with enforced continuity at interfaces
Advantage: Parallelization and better representation capacity than vanilla PINNs
When to use: Arbitrary complex geometries, conservation laws (Wikipedia, 2025)
Distributed PINNs (DPINNs)
Purpose: Solve problems with strong nonlinearity or sharp gradients
How: Space-time domain discretization with lightweight PINNs in larger discrete subdomains
Advantage: Increases accuracy substantially while decreasing computational load
Related: DPIELM (Distributed Physics-Informed Extreme Learning Machines) offers extremely fast lightweight approximation (Wikipedia, 2025)
Bayesian PINNs (B-PINNs)
Purpose: Uncertainty quantification
How: Uses Bayesian framework to provide probability distributions over predictions instead of single values
Methods: Hamiltonian Monte Carlo (HMC) or Variational Inference
Advantage: Provides confidence intervals, robust to noisy data
Limitation: Computationally intensive (Medium, December 2024)
Variational PINNs (VPINNs)
Purpose: Use weak formulations of PDEs
How: Incorporates variational or weak form of PDEs into loss function
Advantage: Handles interface problems, singular solutions, non-smooth geometries
When to use: Problems without classical solutions, only variational forms available (NVIDIA PhysicsNeMo Documentation, 2025)
Physics-Informed PointNet (PIPN)
Purpose: Handle multiple geometries simultaneously
How: Combines PINN loss with PointNet architecture for 3D geometry learning
Advantage: Single training solves for multiple irregular geometries—no retraining needed
Industry impact: Dramatically reduces costs for design parameter investigations (Wikipedia, 2025)
Fourier Feature PINNs
Purpose: Address spectral bias—inability to learn high-frequency functions
How: Uses random Fourier feature embeddings for input coordinates
Advantage: Captures multi-scale and high-frequency phenomena more effectively
Introduced: Wang et al. (2021) in Journal of Computational Physics (AI Review, 2025)
Software Tools and Frameworks
Open-Source Frameworks
DeepXDE
Python library supporting TensorFlow, PyTorch, and JAX backends. Comprehensive implementations of various PINN methods. Well-documented with many examples.
NVIDIA PhysicsNeMo (formerly Modulus)
Professional-grade open-source framework for building, training, and fine-tuning physics AI models at scale. Includes curated architectures like Fourier neural operators, graph neural networks, and diffusion models. GPU-accelerated distributed training. Renamed from Modulus in 2024.
Installation: pip install nvidia-physicsnemo
Key Features: End-to-end pipeline from geometry ingestion to PDE solving, reference applications, multi-GPU support (NVIDIA Developer, 2025).
SciANN
Keras-based framework designed for PINNs. User-friendly for those familiar with Keras.
JAX-based Implementations
Leverage JAX's automatic differentiation and JIT compilation for high performance.
Commercial and Cloud Platforms
MATLAB Deep Learning Toolbox
Built-in PINN support added in recent releases. Integrates with Simulink for system-level modeling. Provides examples for ODEs and PDEs (MATLAB, 2025).
Google Cloud AI Platform
Cloud-based machine learning infrastructure supporting PINN training at scale.
AWS SageMaker
Fully managed service for building and training ML models including PINNs.
Microsoft Azure Machine Learning
Enterprise ML platform with support for custom PINN implementations.
Challenges and Limitations
Despite tremendous promise, PINNs face significant hurdles that researchers are actively addressing.
Computational Cost
Training PINNs is expensive. A study in Journal of Computational Physics (2024) noted that training iterations far exceed those needed for typical data-driven deep neural networks.
The Problem: The physics loss requires evaluating the PDE residual at many collocation points. This involves computing derivatives through automatic differentiation at each training step, which is computationally intensive.
Impact: For simple 1D problems, training might take minutes. For complex 3D fluid dynamics, training can take days on powerful GPUs.
Emerging Solutions: Adaptive sampling strategies, better optimizers, and hierarchical approaches are showing promise. Tensor-based neural networks have demonstrated two orders of magnitude speedup (arXiv, 2024).
Optimization Difficulties
The loss landscape for PINNs is notoriously complex. Multiple competing loss terms can lead to slow convergence or getting stuck in poor local minima.
Spectral Bias: Standard fully-connected networks struggle to learn high-frequency functions—a problem for many physics problems with rapid oscillations (AI Review, 2025).
Multi-Objective Balancing: Choosing weights for data loss, physics loss, and boundary condition loss requires careful tuning. Poor choices lead to networks that fit data but violate physics, or satisfy physics but ignore data.
Solutions Being Explored: Loss-attentional networks that dynamically weight different loss components, Fourier feature embeddings to address spectral bias, and novel activation functions like adaptive activations (ScienceDirect, 2024).
Accuracy Limitations
Current PINNs typically don't match the accuracy of established high-order numerical methods for many problems.
A 2024 study in Advances in Water Resources noted that PINNs modeling flow in heterogeneous porous media face convergence challenges due to property discontinuities. The vanilla PINN structure encounters challenges accurately approximating solutions in regions with "stiffness"—fast-paced alterations in timescale (ResearchGate, 2024).
Scalability Issues
High Dimensionality: While PINNs theoretically handle high-dimensional problems better than grid-based methods, computational requirements still grow rapidly.
Large Datasets: When physics-informed loss requires many collocation points, memory and computation become bottlenecks (MDPI, 2024).
Generalization Concerns
Ensuring PINNs perform well on unseen or out-of-distribution data remains a major research challenge. A trained PINN might work beautifully on the training domain but fail when applied to slightly different conditions (Medium, 2024).
Data Quality Dependencies
PINNs merge data-driven learning with physics but still face challenges with data availability, quality, and diversity. Noisy or biased data can mislead training despite physics constraints (MDPI, 2024).
Current State and Market Context
The Broader Machine Learning Landscape
To understand PINNs' trajectory, we need context from the overall machine learning market:
2024 Market Size: The global machine learning industry is valued at approximately $69-79 billion (multiple sources cite figures in this range as of 2024)
Growth Rate: 30-36% CAGR projected through 2030
2030 Projection: Market expected to reach $500 billion to $1.4 trillion depending on methodology (Precedence Research, MRFR, 2025)
PINNs Within Scientific ML
While specific PINN market figures aren't separately reported, scientific machine learning (SciML) is a rapidly growing niche. Indicators of adoption:
Academic Growth: A Google Scholar survey shows exponential growth in PINN publications from 2019-2024, with thousands of papers published annually across diverse scientific domains (arXiv, 2025).
Industry Adoption: Major technology companies are investing. NVIDIA's commitment to PhysicsNeMo platform signals serious commercial interest. SAP identified PINNs as a key AI theme for 2025, specifically for robotics and predictions grounded in physical reality (SAP News Center, 2025).
Gartner Recognition: PINNs appeared in the Gartner Hype Cycle for Emerging Technologies in 2021, indicating mainstream technology trajectory (Medium, 2024).
Regional Development
North America: Leads in PINN research and commercial deployment. Presence of Brown University (birthplace of PINNs), UC Riverside, University of Pennsylvania, and companies like NVIDIA drive innovation.
Europe: Strong academic research particularly in computational physics. The UK, Germany, and Netherlands have active PINN research groups.
Asia Pacific: Rapidly growing interest, particularly in China and India with large ML workforces.
Sector-Specific Adoption
Manufacturing: 18.88% of global ML market—PINNs seeing use in design optimization and process control
Finance: 15.42% of ML market—emerging applications in option pricing (Medium, February 2025)
Healthcare: Significant adoption for cardiovascular modeling and medical imaging
Aerospace: Active deployment for fluid dynamics and structural analysis
Energy: Wind, solar, and geothermal applications growing rapidly
Investment and Funding
While PINN-specific funding data is limited, the broader AI/ML investment landscape provides context:
2024 Global AI Investment: $252.3 billion, up 44.5% in private investment year-over-year
Key Investors: DARPA invested $2 billion in AI technologies including ML (Fortune Business Insights, 2025)
Enterprise Adoption: 42% of enterprise-scale companies actively use AI, with an additional 40% exploring it
Future Outlook
Near-Term (2025-2027)
Improved Algorithms: Expect continued algorithmic innovations addressing convergence and accuracy issues. Adaptive sampling, attention mechanisms, and better optimization strategies will mature.
Standardization: Community benchmarks and unified evaluation frameworks will emerge. The PINNACLE benchmark introduced in 2023 is an early example (arXiv, 2023).
Hybrid Approaches: Combining PINNs with traditional numerical methods will become standard practice, leveraging strengths of each.
Edge Deployment: Trained PINNs deployed on edge devices for real-time inference in robotics, autonomous vehicles, and IoT applications.
Medium-Term (2027-2030)
Foundation Models: Large pre-trained physics foundation models that can be fine-tuned for specific applications—similar to how GPT works for language.
Multi-Physics Integration: Better handling of coupled physics problems (thermal-mechanical, fluid-structure interaction, electromagnetic-thermal).
Uncertainty Quantification: Bayesian and ensemble methods will mature, providing reliable confidence estimates alongside predictions.
Industry Mainstreaming: PINNs will become standard tools in engineering software alongside FEM and CFD. CAD packages will integrate PINN-based analysis.
Long-Term (2030+)
Automated Discovery: PINNs that can automatically discover unknown governing equations from observational data, accelerating scientific discovery.
Quantum Enhancement: Hybrid quantum-classical PINNs leveraging quantum computing for certain calculations.
Personalized Medicine: Patient-specific PINNs for cardiovascular, cancer, and neurological modeling enabling truly personalized treatment.
Climate Modeling: High-resolution climate models using PINNs to capture regional phenomena with global context.
Research Frontiers
Theory Development: Better mathematical understanding of when and why PINNs work, convergence guarantees, and error bounds.
Architecture Innovation: Novel neural network architectures specifically designed for physics problems. Kolmogorov-Arnold Networks (KANs) showing promise (AI Review, 2025).
Interdisciplinary Expansion: PINNs moving into economics, social sciences, and epidemiology where governing dynamics can be partially modeled.
Myths vs Facts
Myth 1: PINNs Always Beat Traditional Methods
Reality: PINNs excel in specific scenarios—sparse data, inverse problems, complex geometries—but often don't match the accuracy of mature numerical methods for standard forward problems. They're a complementary tool, not a replacement.
Myth 2: PINNs Don't Need Any Data
Reality: While PINNs work with less data than pure ML, they still benefit from measurements. The physics constraints reduce data requirements but don't eliminate them entirely. Hybrid approaches combining physics and data work best.
Myth 3: PINNs Are Always Faster
Reality: Training PINNs can be slower than solving the same problem with traditional numerical methods. The advantage comes during inference and for parametric studies—once trained, querying the network is fast and you don't need to re-mesh for new geometries.
Myth 4: You Need to Be a Physics Expert
Reality: You do need to know the governing equations for your problem. But you don't need to be an expert in numerical methods or mesh generation—that's precisely what PINNs eliminate.
Myth 5: PINNs Only Work for Simple Problems
Reality: Early demonstrations focused on canonical problems, but recent work tackles industrial-scale applications. Shell's chemical reactors, SoftServe's oil well monitoring, and aerospace fluid dynamics prove PINNs handle real-world complexity.
Myth 6: All PINNs Are the Same
Reality: "PINN" has become an umbrella term covering many variants—XPINNs, VPINNs, Bayesian PINNs, each with different strengths. Choosing the right variant for your problem is important.
Getting Started: Practical Checklist
If you want to explore PINNs for your problem, here's a roadmap:
Assessment Phase
[ ] Define Your Problem: Do you have a differential equation describing your system? If not, PINNs may not be the right tool.
[ ] Evaluate Data Availability: PINNs shine with sparse data. If you have abundant data, pure ML might suffice. If you have no data, traditional numerical methods might be better.
[ ] Check Geometry Complexity: PINNs excel with irregular or changing geometries.
[ ] Consider Inverse Problem Needs: Estimating parameters from observations? PINNs are excellent for this.
Learning Phase
[ ] Study Fundamentals: Understand neural networks, automatic differentiation, and your specific PDEs.
[ ] Review Literature: Read the original 2019 Raissi et al. paper in Journal of Computational Physics.
[ ] Explore Examples: Work through tutorials in DeepXDE or NVIDIA PhysicsNeMo documentation.
Implementation Phase
[ ] Choose Framework: Start with DeepXDE for ease of use or PhysicsNeMo for scalability.
[ ] Start Simple: Begin with 1D or 2D problems to understand behavior.
[ ] Implement Loss Function: Carefully construct data, physics, and boundary losses.
[ ] Tune Hyperparameters: Experiment with network architecture, learning rates, and loss weights.
Validation Phase
[ ] Validate Against Known Solutions: Test on problems with analytical solutions first.
[ ] Assess Convergence: Monitor all loss components during training.
[ ] Check Physics Compliance: Verify that predictions actually satisfy governing equations.
[ ] Compare with Baselines: Benchmark against traditional numerical methods or pure ML.
Deployment Phase
[ ] Optimize for Inference: Once trained, optimize the network for fast deployment.
[ ] Document Assumptions: Clearly state the physical assumptions and valid parameter ranges.
[ ] Plan for Updates: Establish protocols for retraining with new data.
FAQ
Q1: Can PINNs work without any training data?
Yes, in theory. If you have complete physics (governing equations, boundary conditions, initial conditions), PINNs can solve the forward problem without measurements. However, some validation data is always recommended, and hybrid approaches with at least sparse measurements typically perform better.
Q2: How long does it take to train a PINN?
Training time varies dramatically. Simple 1D problems might train in minutes on a CPU. Complex 3D fluid dynamics problems can take days on high-end GPUs. As a rule of thumb, expect PINN training to be slower than solving the same problem once with traditional numerical methods, but faster than running hundreds of parametric studies.
Q3: What programming languages do I need?
Python dominates PINN research and application. Frameworks like DeepXDE (TensorFlow/PyTorch), NVIDIA PhysicsNeMo, and JAX-based implementations all use Python. MATLAB also has built-in PINN capabilities in its Deep Learning Toolbox.
Q4: Can PINNs handle turbulence?
Turbulence is challenging for vanilla PINNs due to multi-scale behavior and strong nonlinearities. However, specialized approaches like Distributed PINNs (DPINNs) and hybrid methods combining PINNs with turbulence models show promising results. Zhang et al. (2024) successfully applied PINNs with k-ε turbulence models.
Q5: What hardware do I need?
For learning and simple problems, a standard CPU suffices. For research and complex problems, a GPU with at least 8GB memory is recommended. Multi-GPU setups dramatically speed up training for large-scale problems. Cloud platforms (AWS, Google Cloud, Azure) offer on-demand GPU resources.
Q6: How do PINNs compare in cost to traditional simulation?
The economics depend on your use case. Traditional methods have lower upfront cost for single simulations but higher cost for parametric studies requiring many runs. PINNs have high training cost but very low inference cost, making them economical when you need to evaluate many designs or real-time predictions.
Q7: Can PINNs discover unknown physics?
Yes, this is called the inverse problem. PINNs can estimate unknown parameters in differential equations from observational data. They can also help identify which terms in an equation are important. However, discovering completely unknown equations requires hybrid approaches with symbolic regression.
Q8: What types of differential equations can PINNs handle?
PINNs work with ordinary differential equations (ODEs), partial differential equations (PDEs), integro-differential equations, fractional equations, and stochastic PDEs. They handle linear and nonlinear equations, time-dependent and steady-state problems.
Q9: Do PINNs replace engineers?
No. PINNs are tools that augment engineering workflows, not replacements for human expertise. Engineers still need to formulate problems, choose appropriate physics models, interpret results, and make design decisions. PINNs eliminate tedious meshing and enable faster iteration.
Q10: What's the biggest challenge in using PINNs?
The biggest practical challenge is balancing the multiple loss terms and achieving reliable convergence. This requires experimentation and domain knowledge. The biggest theoretical challenge is achieving accuracy competitive with mature numerical methods for all problem types.
Q11: Can I use PINNs for real-time applications?
Once trained, PINNs offer fast inference suitable for real-time applications. Examples include the SoftServe oil well monitoring and Kinetic Vision design optimization deployed with real-time feedback. However, training remains offline and computationally intensive.
Q12: How do I know if PINNs are right for my problem?
PINNs are a good fit if you have: 1) governing equations describing your system, 2) sparse or expensive data, 3) complex geometries that make meshing difficult, 4) need for inverse problem solution, or 5) requirement for many parametric evaluations. They're less suitable if you have abundant data but no physics model, or if you need maximum accuracy for a single forward simulation.
Key Takeaways
PINNs embed physical laws into neural networks, enabling accurate predictions with sparse data by using differential equations as training constraints.
Introduced in 2017 by Raissi, Perdikaris, and Karniadakis, PINNs have rapidly grown into a major pillar of scientific machine learning with over 30,000 citations.
Real industrial applications exist today—Shell achieves 100 million times faster reactor simulations, SoftServe monitors oil wells, and Kinetic Vision optimizes designs with real-time feedback.
Key advantages include meshfree operation, handling complex geometries, excelling at inverse problems, and working with limited data.
Current limitations include high computational training cost, optimization challenges, accuracy gaps compared to mature numerical methods, and convergence issues for complex problems.
Many specialized variants address specific needs: XPINNs for domain decomposition, Bayesian PINNs for uncertainty, Variational PINNs for weak formulations, and Fourier PINNs for high-frequency phenomena.
Professional tools are available including open-source frameworks (DeepXDE, NVIDIA PhysicsNeMo) and commercial platforms (MATLAB, AWS, Google Cloud, Azure).
The field is rapidly maturing with algorithmic improvements, better optimization strategies, and growing industry adoption across aerospace, healthcare, energy, and manufacturing.
PINNs complement rather than replace traditional methods, excelling in scenarios with sparse data, complex geometries, or parametric studies while established numerical methods remain superior for standard forward simulations.
Future outlook is strong with foundation models, multi-physics integration, edge deployment, and mainstream engineering software integration expected by 2030.
Actionable Next Steps
Learn the Fundamentals: Read the original 2019 Raissi et al. paper "Physics-informed neural networks: A deep learning framework" in Journal of Computational Physics to understand core concepts.
Install a Framework: Download DeepXDE (pip install deepxde) for beginners or NVIDIA PhysicsNeMo (pip install nvidia-physicsnemo) for scalability. Work through official tutorials.
Start with a Toy Problem: Implement a PINN for a simple 1D ODE or heat equation. This teaches workflow without complexity.
Identify a Real Problem in Your Domain: Look for scenarios with sparse data, expensive simulations, or complex geometries where PINNs might add value.
Join the Community: Follow research groups at Brown University (Karniadakis), UC Riverside (Raissi), University of Pennsylvania (Perdikaris). Engage in forums and GitHub repositories.
Benchmark Carefully: When applying PINNs to real problems, always compare against traditional methods and validate physics compliance, not just data fitting.
Experiment with Variants: If vanilla PINNs struggle with your problem, try XPINNs for domain decomposition, Variational PINNs for weak forms, or Fourier features for high-frequency phenomena.
Consider Hybrid Approaches: Combine PINNs with traditional numerical methods, using each where they perform best.
Monitor the Literature: The field evolves rapidly. Follow recent papers on arXiv, Journal of Computational Physics, and AI conferences.
Pilot Before Production: Start with proof-of-concept projects to build expertise before deploying PINNs in critical applications.
Glossary
Automatic Differentiation: Technique for computing derivatives of functions specified by computer programs, essential for training PINNs.
Bayesian PINN (B-PINN): Variant using Bayesian framework to quantify uncertainty in predictions by providing probability distributions rather than single values.
Boundary Conditions: Constraints on solution values at domain boundaries, e.g., fixed temperature at a wall.
Collocation Points: Sample points in the domain where the PDE residual is evaluated during PINN training.
Deep Neural Network (DNN): Neural network with multiple hidden layers enabling learning of complex patterns.
Distributed PINN (DPINN): Variant decomposing space-time domain into subdomains solved by separate lightweight PINNs.
Extended PINN (XPINN): Variant using domain decomposition with multiple neural networks for complex geometries.
Finite Element Method (FEM): Traditional numerical technique dividing domain into mesh of elements.
Forward Problem: Predicting system behavior given governing equations and parameters.
Fourier Neural Operator (FNO): Neural network architecture operating in Fourier space, often combined with PINN approaches.
Inverse Problem: Estimating unknown parameters in governing equations from observational data.
Loss Function: Mathematical function measuring how well a neural network's predictions match desired outcomes.
Meshfree Method: Numerical approach not requiring domain discretization into a mesh.
Navier-Stokes Equations: Fundamental PDEs describing fluid motion.
Ordinary Differential Equation (ODE): Equation involving derivatives with respect to a single variable.
Partial Differential Equation (PDE): Equation involving partial derivatives with respect to multiple variables.
Physics-Informed Neural Network (PINN): Neural network incorporating physical laws described by differential equations into training.
Residual: Measure of how well a solution satisfies a differential equation; PINNs minimize this.
Scientific Machine Learning (SciML): Field combining machine learning with scientific computing and physics.
Spectral Bias: Tendency of standard neural networks to learn low-frequency functions easily but struggle with high-frequency components.
Surrogate Model: Simplified model approximating complex simulations for faster evaluation.
Transfer Learning: Using knowledge from one problem to accelerate learning on related problems.
Universal Function Approximator: Mathematical object (like neural networks) capable of approximating any continuous function given sufficient parameters.
Variational PINN (VPINN): Variant using weak formulation of PDEs, suitable for problems without classical solutions.
Sources and References
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686-707. https://www.sciencedirect.com/science/article/abs/pii/S0021999118307125
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2017). Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations. arXiv preprint arXiv:1711.10561. https://arxiv.org/abs/1711.10561
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2017). Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations. arXiv preprint arXiv:1711.10566. https://arxiv.org/abs/1711.10566
Physics-informed neural networks. (2025, October). Wikipedia. https://en.wikipedia.org/wiki/Physics-informed_neural_networks
What Are Physics-Informed Neural Networks (PINNs)? (2025). MATLAB & Simulink. MathWorks. https://www.mathworks.com/discovery/physics-informed-neural-networks.html
Cuomo, S., Di Cola, V. S., Giampaolo, F., Rozza, G., Raissi, M., & Piccialli, F. (2022). Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What's Next. Journal of Scientific Computing, 92(3). https://link.springer.com/article/10.1007/s10915-022-01939-z
Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics, 3(6), 422-440.
Cai, S., Mao, Z., Wang, Z., Yin, M., & Karniadakis, G. E. (2021). Physics-informed neural networks (PINNs) for fluid mechanics: A review. Acta Mechanica Sinica, 37(12), 1727-1738.
Luo, R., et al. (2025). Physics-informed neural networks for PDE problems: a comprehensive review. Artificial Intelligence Review. Springer. https://link.springer.com/article/10.1007/s10462-025-11322-7
Tkachuk, P., Krasikov, M., & Kasianiuk, V. (2025). Physics-Informed Neural Networks in Aerospace. Challenges and Issues of Modern Science, 4(1). https://philarchive.org/archive/TKAPNN
Dagrada, M. (2024, December). Introduction to Physics-informed Neural Networks. Medium - TDS Archive. https://medium.com/data-science/solving-differential-equations-with-neural-networks-afdcf7b8bcc4
Physics Informed Neural Networks (PINNs) in PhysicsNeMo Sym. (2025). NVIDIA PhysicsNeMo Framework. https://docs.nvidia.com/physicsnemo/latest/physicsnemo-sym/user_guide/theory/phys_informed.html
Huang, B., & Wang, J. (2022). Applications of physics-informed neural networks in power systems—A review. IEEE Transactions on Power Systems, 38(1), 572-588.
Feng, Y., Eun, J., Kim, S., & Kim, Y. (2025). Application of physics-informed neural networks (PINNs) solution to coupled thermal and hydraulic processes in silty sands. International Journal of Geo-Engineering, 16(1). https://www.osti.gov/pages/biblio/2500990
Abdelsattar, M., et al. (2025). Integrating data-driven and physics-based approaches for robust wind power prediction: A comprehensive ML-PINN-Simulink framework. Scientific Reports. Nature. https://www.nature.com/articles/s41598-025-13306-7
Malik, H., et al. (2024). Advancements in Physics-Informed Neural Networks for Laminated Composites: A Comprehensive Review. Mathematics, 13(1), 17. MDPI. https://www.mdpi.com/2227-7390/13/1/17
Sahli Costabal, F., et al. (2020). Physics-informed neural networks for cardiac activation mapping. Frontiers in Physics, 8, 42.
Kissas, G., et al. (2020). Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 358, 112623.
Eidell, M. (2022, August). Accelerating Product Development with Physics-Informed Neural Networks and NVIDIA PhysicsNeMo. NVIDIA Technical Blog. https://developer.nvidia.com/blog/accelerating-product-development-with-physics-informed-neural-networks-and-modulus/
Exploring the Potential of Physics-Informed Neural Networks for AI Applications. (2024, March). SoftServe Blog. https://www.softserveinc.com/en-us/blog/exploring-the-potential-of-physics-informed-neural
AI in 2025: 5 Themes. (2025, January). SAP News Center. https://news.sap.com/2025/01/ai-in-2025-defining-themes/
Precedence Research. (2025). Machine Learning Market Size to Worth USD 1,407.65 Bn By 2034. https://www.precedenceresearch.com/machine-learning-market
Machine Learning Statistics 2024. AIPRM. https://www.aiprm.com/machine-learning-statistics/
The Ultimate List of Machine Learning Statistics for 2025. ITTransition. https://www.itransition.com/machine-learning/statistics
Machine Learning Market Size & Share | Industry Report 2030. Grand View Research. https://www.grandviewresearch.com/industry-analysis/machine-learning-market
70+ Machine Learning Statistics 2025: Industry Market Size. DemandSage. https://www.demandsage.com/machine-learning-statistics/
Fortune Business Insights. Machine Learning Market Size, Share, Growth | Trends [2032]. https://www.fortunebusinessinsights.com/machine-learning-market-102226
Market Research Future. Machine Learning Market Size, Growth Analysis, 2032. https://www.marketresearchfuture.com/reports/machine-learning-market-2494
AI Statistics 2024–2025: Global Trends, Market Growth & Adoption Data. Founders Forum Group. https://ff.co/ai-statistics-trends-global-market/
NVIDIA PhysicsNeMo. (2025). NVIDIA Developer. https://developer.nvidia.com/modulus
NVIDIA/modulus GitHub Repository. (2025). https://github.com/NVIDIA/modulus
Raissi, M. (2025). Authors | Physics Informed Deep Learning. https://maziarraissi.github.io/PINNs/
Maziar Raissi Google Scholar Profile. https://scholar.google.com/citations?user=dCdmUaYAAAAJ
Paris Perdikaris Google Scholar Profile. https://scholar.google.com/citations?user=h_zkt1oAAAAJ
Physics-informed neural networks for physiological signal processing and modeling: a narrative review. (2024). PMC. https://pmc.ncbi.nlm.nih.gov/articles/PMC12308510/
Adapting physics-informed neural networks to improve ODE optimization in mosquito population dynamics. (2024). PLOS ONE, 19(12), e0315762. https://pmc.ncbi.nlm.nih.gov/articles/PMC11666042/
Faroughi, S. A., et al. (2024). Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks and Operators in Scientific Computing: Fluid and Solid Mechanics. ASME Journal of Computing and Information Science in Engineering, 24(4), 040802. https://asmedigitalcollection.asme.org/computingengineering/article/24/4/040802/1193884
Understanding Physics-Informed Neural Networks: Techniques, Applications, Trends, and Challenges. (2024). AI, 5(3), 1534-1557. MDPI. https://www.mdpi.com/2673-2688/5/3/74
A comprehensive analysis of PINNs: Variants, Applications, and Challenges. (2025). arXiv preprint arXiv:2505.22761v1. https://arxiv.org/html/2505.22761v1
Grigoryan, A. A. (2024, December). Advancing Physics-Informed Neural Networks (PINNs): Their Role, Extensions, and Challenges — Part 3. Medium. https://thegrigorian.medium.com/advancing-physics-informed-neural-networks-pinns-their-role-extensions-and-challenges-part-3-dee1baa28a1a
Physics Informed Neural Networks (PINNs): An Intuitive Guide. (2025, January). Towards Data Science. https://towardsdatascience.com/physics-informed-neural-networks-pinns-an-intuitive-guide-fff138069563
Full article: Physics-Informed neural network solver for numerical analysis in geoengineering. (2024). Taylor & Francis Online. https://www.tandfonline.com/doi/full/10.1080/17499518.2024.2315301
Raissi, M., Perdikaris, P., Ahmadi, N., & Karniadakis, G. E. (2024). Physics-Informed Neural Networks and Extensions. arXiv preprint arXiv:2408.16806. https://arxiv.org/abs/2408.16806
Evolutionary Optimization of Physics-Informed Neural Networks: Survey and Prospects. (2025). arXiv preprint arXiv:2501.06572v2. https://arxiv.org/html/2501.06572v2
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