What is Physics-Informed Machine Learning (PIML)?
- Muiz As-Siddeeqi
- Oct 10
- 19 min read
Imagine training an artificial intelligence that doesn't just learn from data—it understands the laws of physics. That's the breathtaking promise of Physics-Informed Machine Learning, a revolutionary approach that's transforming how we predict weather patterns, discover life-saving drugs, and solve complex engineering problems. While traditional AI devours millions of data points to make predictions, PIML achieves remarkable accuracy with a fraction of the information by incorporating centuries of scientific knowledge directly into its neural networks.
TL;DR
PIML integrates physical laws (like gravity, thermodynamics, or fluid dynamics) directly into machine learning models, making them smarter and more efficient
Drastically reduces data requirements—often by orders of magnitude—compared to purely data-driven approaches
Key method is PINNs (Physics-Informed Neural Networks), introduced in 2017 and now used across dozens of scientific domains
Real-world impact: accelerating drug discovery, improving weather forecasts, optimizing building energy use, and advancing autonomous systems
Major challenges: computational complexity, implementation difficulty, and ensuring models generalize across different scenarios
Physics-Informed Machine Learning (PIML) is a machine learning approach that embeds established physical laws—such as conservation of mass, energy, or momentum—directly into AI algorithms. By constraining models with scientific principles expressed as mathematical equations, PIML achieves superior accuracy, generalization, and data efficiency compared to traditional data-driven methods, particularly in scientific and engineering applications where physical consistency is critical.
Table of Contents
What is Physics-Informed Machine Learning?
Physics-Informed Machine Learning represents a paradigm shift in artificial intelligence. Instead of relying solely on data patterns, PIML incorporates fundamental physical laws—conservation principles, partial differential equations (PDEs), symmetries, and invariances—directly into machine learning algorithms.
Traditional machine learning operates like a student who memorizes answers without understanding principles. PIML, by contrast, embeds the rulebook itself into the learning process. When predicting fluid flow, for example, a PIML model doesn't just learn from millions of simulation snapshots—it respects the Navier-Stokes equations that govern all fluid behavior.
According to a comprehensive review published in Nature Reviews Physics (Karniadakis et al., 2021), PIML "integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks."
Why PIML Matters
The significance of PIML becomes clear when you consider its advantages:
Data Efficiency: A study in Machine Learning for Computational Science and Engineering (2025) notes that PIML models can reduce training data requirements by several orders of magnitude compared to purely data-driven approaches.
Physical Consistency: Unlike black-box models that can produce physically impossible predictions, PIML ensures outputs obey fundamental laws. A review in Expert Systems with Applications (July 2024) emphasizes that PIML models maintain "physical plausibility of results."
Generalization: PIML models perform better on scenarios they've never seen. Research published in Applied Energy (March 2025) demonstrates that physics-informed models show "enhanced robustness and interpretability" when applied to new conditions.
The Historical Context: From Traditional Models to PIML
The Pre-PIML Era
For centuries, scientists solved physical problems using two approaches:
Physics-Based Numerical Methods (1940s-2000s): Engineers discretized equations using techniques like Finite Element Method (FEM) or Finite Difference Method (FDM). These approaches were accurate but computationally expensive and required extensive domain expertise.
Pure Data-Driven Machine Learning (2010s): With the deep learning revolution, researchers began using neural networks for scientific problems. However, these models demanded massive datasets and often produced physically inconsistent results.
The Birth of PINNs (2017)
The modern PIML era began in 2017 when Maziar Raissi, Paris Perdikaris, and George Em Karniadakis introduced Physics-Informed Neural Networks (PINNs) in a groundbreaking paper. Their work, published in the Journal of Computational Physics (2019), demonstrated that neural networks could be trained to solve partial differential equations while respecting physical laws.
A 2022 review in the Journal of Scientific Computing notes that PINNs arose "as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual."
Rapid Growth (2018-Present)
According to data from Philosophical Transactions of the Royal Society A (April 2021), the field has exploded. A geotechnical engineering review published in Geotechnical Engineering (2025) shows publications on PIML in soil and rock mechanics grew from nearly zero before 2019 to hundreds by 2024.
How PIML Works: Core Concepts
The Three Pillars
PIML stands on three foundational elements:
1. Observational Data: Real-world measurements (temperature readings, sensor data, experimental results)
2. Physical Laws: Mathematical expressions of natural phenomena (Newton's laws, thermodynamic equations, conservation principles)
3. Machine Learning Architecture: Neural networks or other ML models that can learn from both data and constraints
The Integration Process
A survey in Machine Learning for Computational Science and Engineering (May 2025) identifies three main integration approaches:
Loss Function Modification: The most common method adds physics-based penalty terms to the standard loss function. When the model violates physical laws, it incurs a higher loss.
Architectural Design: Some approaches build physics directly into the network structure. For instance, designing networks that automatically conserve energy or momentum.
Feature Engineering: Incorporating physics-derived features as inputs or using physics to guide data preprocessing.
Mathematical Foundation
At its core, PIML solves problems where the solution must satisfy:
Data fitting: Minimize error between predictions and observations
Physics constraints: Satisfy governing equations (often PDEs)
Boundary/Initial conditions: Respect problem-specific constraints
For example, when modeling heat distribution, a PIML model simultaneously:
Fits to temperature sensor data
Obeys the heat equation (∂T/∂t = α∇²T)
Respects boundary temperatures at walls
What Makes PINNs Special
PINNs are the most prominent PIML technique. According to a comprehensive review published in AI (August 2024), PINNs "encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself."
How PINNs Work
Training Process:
Sample points from the problem domain (no mesh required!)
Feed points through neural network to get predictions
Use automatic differentiation to compute derivatives
Calculate residuals of the governing PDE
Minimize combined loss: data error + PDE residual + boundary condition error
Key Innovation: Automatic differentiation allows computing derivatives without manually coding complex mathematics.
PINN Variants
Since 2017, researchers have developed numerous PINN variations:
XPINNs (Extended PINNs): Decompose problems into subdomains for parallel computation, improving scalability for complex geometries
VPINNs (Variational PINNs): Use weak formulations of PDEs for better accuracy with higher-order equations
Bayesian PINNs: Quantify uncertainty in predictions, critical for safety-critical applications
PIKANs (Physics-Informed Kolmogorov-Arnold Networks): A 2024 development leveraging alternative neural network architectures based on Kolmogorov's 1957 representation theorem
An October 2024 review paper titled From PINNs to PIKANs (arXiv:2410.13228) notes that PINNs have evolved significantly, with improvements in "network architectures, adaptive refinement, domain decomposition, and the use of adaptive weights and activation functions."
Methods of Integrating Physics into Machine Learning
1. Physics-Informed Loss Functions
The most widely adopted approach. A review in Journal of Manufacturing Processes (January 2025) categorizes this as "physics-based modification of the loss function."
How it works: Add terms to the loss function that penalize violations of physical laws.
Example: For fluid dynamics, the loss includes:
Data fit term: ||u_predicted - u_observed||²
Physics term: ||∂u/∂t + u·∇u + ∇p - ν∇²u||² (Navier-Stokes residual)
Boundary term: ||u(boundary) - u_BC||²
Advantage: Flexible and easy to implement with existing neural network frameworks
Limitation: Requires careful weight balancing between different loss terms
2. Physics-Based Architecture Design
Building physical constraints directly into network structure. Research in Applied Energy (March 2025) describes this as "physics-informed architectural design."
Examples:
Hamiltonian Neural Networks: Automatically conserve energy
Symplectic networks: Preserve geometric structure of physical systems
Equivariant networks: Respect symmetries (rotation, translation)
Advantage: Physical laws are enforced by construction, not just encouraged
Limitation: Requires deep understanding of both physics and network architecture
3. Physics-Based Feature Engineering
Using domain knowledge to construct informative features. The Journal of Manufacturing Processes review calls this "physics-based feature engineering."
Approach: Instead of feeding raw data, transform inputs using physics-derived quantities (Reynolds number, Prandtl number, dimensionless groups).
Advantage: Reduces learning complexity and improves generalization
Limitation: Requires extensive domain expertise upfront
4. Hybrid Approaches
Combining multiple methods. A ScienceDirect review (November 2024) notes increasing interest in "hybrid frameworks that combine neural networks with simplified physical models."
Real-World Applications
PIML has penetrated numerous industries. Here's where it's making the biggest impact:
Weather and Climate Science
PIML is revolutionizing atmospheric modeling. The European Centre for Medium-Range Weather Forecasts (ECMWF) reported in 2024 that ML-based predictions can match or exceed traditional numerical weather prediction for 90% of variables while generating forecasts in seconds instead of hours.
Drug Discovery and Healthcare
Google DeepMind's AlphaFold 3, released in May 2024, uses physics-informed approaches to predict protein structures and interactions. The system has predicted over 200 million protein structures—nearly all catalogued proteins known to science.
Energy and Buildings
Research published in Applied Energy (March 2025) shows PIML models for building energy management reduce prediction errors while requiring 60-80% less training data than pure ML approaches.
Manufacturing and Materials
A review in the Journal of Manufacturing Processes (January 2025) demonstrates PIML's value in additive manufacturing, where it models process-structure-property relationships with "enhanced robustness and interpretability."
Aerospace Engineering
Studies collected in a 2024 aerospace review show PINNs successfully model thermal systems in spacecraft, natural convection in engines, and combustion reaction kinetics.
Nuclear Engineering
A 2023 Scientific Reports paper describes using transfer learning with PINNs (TL-PINNs) to predict nuclear reactor transients, achieving "significant performance gain" by reducing training iterations.
Case Study 1: Weather Forecasting and Climate Modeling
The Challenge
Traditional Numerical Weather Prediction (NWP) systems require massive computational resources. The global models run on supercomputers and take hours to produce forecasts. A 2023 Bulletin of the American Meteorological Society paper noted that NWP faces "inherent atmospheric uncertainties and computational costs, especially in the post-Moore era."
The PIML Solution
Researchers have developed multiple physics-informed weather models:
GraphCast (Google DeepMind, 2023): Uses graph neural networks with physics constraints to predict weather 10 days ahead
FourCastNet (NVIDIA, 2022): Employs Fourier transforms and physical constraints for global weather prediction
Pangu-Weather (Huawei, 2023): 3D neural networks incorporating atmospheric physics
Documented Results
A study published in Philosophical Transactions of the Royal Society A (April 2021) examined 10 PIML case studies in weather and climate modeling, finding:
"Greater physical consistency" in predictions
"Reduced training time" compared to pure data-driven models
"Improved data efficiency"—models trained with less data performed better
"Better generalization" to unseen weather patterns
A January 2025 study in Artificial Intelligence for the Earth Systems compared GraphCast, Pangu-Weather, and FourCastNet against ECMWF's traditional system during three extreme events:
2021 Pacific Northwest heatwave
2023 South Asian humid heatwave
2021 North American winter storm
The physics-informed models achieved competitive accuracy while running 100-1000x faster.
Real-World Impact
According to Atmosphere (June 2024), data-driven weather models "can provide effective forecasts with an accuracy (ACC) greater than 0.6 for up to 15 days at a spatial resolution of 0.25°" and "reduce forecast generation time from hours to seconds."
Case Study 2: Drug Discovery and Protein Folding
The Protein Folding Problem
Determining a protein's 3D structure from its amino acid sequence historically took years and cost hundreds of thousands of dollars. As of 2021, scientists had mapped only 17% of the human body's 20,000 proteins using traditional methods.
AlphaFold: A Physics-Informed Breakthrough
Google DeepMind's AlphaFold series represents one of PIML's most celebrated successes:
AlphaFold 2 (2020): Achieved breakthrough accuracy in protein structure prediction, winning the CASP14 competition. Developers Demis Hassabis and John Jumper received the 2024 Nobel Prize in Chemistry.
AlphaFold 3 (May 2024): Extended predictions to protein-protein interactions, protein-DNA, protein-RNA, and protein-ligand (drug molecule) interactions.
How It Uses Physics
While not a classical PINN, AlphaFold incorporates physical principles:
Geometric constraints from protein chemistry
Energy minimization principles
Known protein folding patterns
Spatial relationship rules from structural biology
Impact on Drug Discovery
A November 2024 analysis in LabioTech reports:
"For the interactions of proteins with other molecule types we see at least a 50% improvement compared with existing prediction methods, and for some important categories of interaction we have doubled prediction accuracy."
AlphaFold has been cited over 20,000 times. The AlphaFold Protein Structure Database has over 2 million users in 190 countries.
Real Applications
Malaria vaccine research: AlphaFold helped clarify protein structures that were previously too unclear to identify effective targets
Cancer drug development: Researchers at the Institute of Cancer Research note that AlphaFold helps "accurately design and discover better, safer drugs"
Speed: A January 2024 study in Freethink quoted researchers stating AlphaFold "could advance the project by a couple of years"
Verification Studies
A preprint study led by researchers at UC San Francisco and UNC Chapel Hill (2024) tested whether AlphaFold's predictions actually help find drug candidates. They identified hundreds of promising compounds and found "hit rates" comparable to traditional structure-based methods—validating AlphaFold's practical utility.
Case Study 3: Building Energy Management
The Problem
Building energy modeling faces conflicting challenges:
Physics-based models: Accurate but require detailed building information, expert knowledge, and case-by-case calibration
Data-driven models: Flexible but "suffer from limited generalization ability and a lack of physical consistency" (ScienceDirect, May 2025)
The PIML Approach
Research published in Applied Energy (March 2025) provides "a comprehensive review of Physics-Informed Machine Learning (PIML) methods for Building Energy Modeling (BEM)."
The study identifies four main PIML paradigms:
Physics-informed inputs: Using physics-derived features (heat transfer coefficients, thermal mass indicators)
Physics-informed loss functions: Penalizing violations of energy conservation or thermodynamic laws
Physics-informed architecture: Networks designed to respect building physics
Physics-informed ensemble models: Combining physics-based and data-driven components
Quantified Benefits
The Applied Energy review documents several advantages:
Data Efficiency: PIML models require "60-80% less training data" than pure ML approaches while maintaining accuracy
Robustness: Physics constraints reduce overfitting, improving performance on buildings not in training data
Interpretability: Unlike black-box models, PIML models' physics components provide transparent reasoning
Physical Consistency: Predictions never violate conservation of energy or thermodynamic principles
Real Deployments
A May 2025 ScienceDirect study examining PIML for building performance simulation notes these models help with:
Optimizing heating, ventilation, and air conditioning (HVAC) systems
Predicting energy consumption for different building designs
Supporting smart building control in real-time
Analyzing building resilience to climate change
Comparison with Traditional Methods
The same study found:
Aspect | Physics-Based | Pure ML | PIML |
Data requirements | Low | Very high | Moderate |
Modeling effort | Very high | Low | Moderate |
Generalization | Good | Poor | Excellent |
Computational cost (training) | Low | High | Moderate |
Computational cost (inference) | High | Very low | Very low |
Physical consistency | Perfect | Poor | Excellent |
Benefits of PIML
1. Dramatic Data Efficiency
The most compelling advantage. A Medium article (March 2025) notes that PIML makes "model training highly data-efficient, i.e. trainable with fewer data."
Quantified improvements:
Pacific Northwest National Laboratory reports PIML can reduce training samples "by several orders of magnitude"
Building energy models need 60-80% less data (Applied Energy, March 2025)
Additive manufacturing models achieve comparable accuracy with 10x less training data (Journal of Manufacturing Processes, January 2025)
Why it matters: For scientific problems where data collection is expensive (climate modeling, nuclear reactor testing, aerospace engineering), this is transformative.
2. Physical Consistency
Unlike pure ML models, PIML outputs always obey fundamental laws.
A review in Expert Systems with Applications (July 2024) emphasizes that PIML provides "physical plausibility of results."
Real examples:
Fluid dynamics models that never violate mass conservation
Climate models that respect energy balance
Structural analysis that honors Newton's laws
3. Superior Generalization
PIML models perform better on conditions they've never seen. The Applied Energy review (March 2025) notes "generalization capabilities of the model, such that models can make better prediction for scenarios unseen during the training phase."
Case in point: Weather models trained on one geographic region can better predict weather in different regions when physics-informed.
4. Faster Training and Convergence
Physics constraints guide models toward correct solutions more quickly. Medium (March 2025) reports "acceleration of the model training process, such that the models converge faster to an optimal solution."
5. Enhanced Interpretability
Unlike black-box neural networks, PIML models offer transparency. The Applied Energy study notes "improvement of transparency and interpretability of models."
Benefits:
Regulators can verify model decisions
Engineers understand failure modes
Scientists gain insights into physical phenomena
6. Mesh-Free Solutions
Traditional numerical methods require tedious mesh generation. A MATLAB explainer notes PINNs "are mesh-free," significantly reducing preprocessing effort.
7. Inverse Problem Solving
PIML excels at estimating unknown parameters from observations. Applications include:
Determining material properties from sensor data
Identifying disease parameters from patient data
Inferring boundary conditions in engineering problems
Challenges and Limitations
Despite its promise, PIML faces significant hurdles.
1. Computational Complexity
Training PINNs can be expensive. A survey in Machine Learning for Computational Science and Engineering (May 2025) identifies "data and computational costs" as a major challenge, noting that "developing PDE foundation models is resource-intensive."
The problem: Computing PDE residuals at many points using automatic differentiation is computationally intensive. A 2024 aerospace study noted that parameterizing three variables in a P-PINN increased computational cost by 46%.
2. Training Instability
PINNs can be difficult to train. A Physics of Fluids review (October 2024) notes challenges with "gradient pathologies" and training convergence.
Common issues:
Multiple loss terms require careful weight balancing
Stiff PDEs cause gradient problems
High-dimensional problems face curse of dimensionality
3. Limited Effectiveness for Unknown Physics
PIML requires knowing the governing equations. When physics is poorly understood or partially known, benefits diminish. The Machine Learning for Computational Science and Engineering survey (May 2025) notes: "PIML's success in mitigating data requirements is more context-specific and is most effective when the underlying physical processes are well-understood."
4. Architecture Design Complexity
Building physics directly into network architectures demands expertise in both domains. Few practitioners possess deep knowledge of both advanced machine learning and domain physics.
5. Scalability Challenges
High-dimensional problems remain difficult. A Nature Reviews Physics paper (May 2021) acknowledges that "high-dimensional problems governed by parameterized PDEs cannot be tackled" easily even with PIML.
6. Lack of Theoretical Guarantees
Unlike traditional numerical methods with known error bounds, PIML offers fewer theoretical guarantees. A review in Energies (February 2023) notes this as a key limitation.
7. Benchmarking Difficulties
Different PIML approaches are hard to compare fairly. The Energies review states: "it is currently not entirely clear whether physics-informed features, labels, architectures, and loss functions outperform competing ML paradigms."
PIML vs Traditional Approaches
Comparison Table
Factor | Traditional Numerical | Pure ML | PIML |
Accuracy | High (for known physics) | Varies widely | High |
Data requirements | Low | Very high | Low to moderate |
Computational cost (solve) | Very high | Low | Low |
Generalization | Good (within physics domain) | Poor (data-dependent) | Excellent |
Interpretability | High | Very low | Moderate to high |
Mesh generation required | Yes | No | No |
Handles unknown physics | No | Yes | Limited |
Training complexity | N/A | Moderate | High |
Real-time capability | Limited | Excellent | Excellent |
Error quantification | Well-established | Difficult | Emerging |
When to Use Each Approach
Traditional Numerical Methods when:
Exact solutions with error bounds are needed
Physics is completely known
One-time simulations (not repeated predictions)
Maximum accuracy is essential
Pure Machine Learning when:
Physics is unknown or extremely complex
Abundant high-quality data available
Interpretability isn't critical
Learning hidden patterns is the goal
PIML when:
Physics is known but complex
Data is limited or expensive
Physical consistency is essential
Fast repeated predictions needed
Generalization beyond training data is critical
Inverse problems must be solved
Industry Adoption and Future Outlook
Current State
PIML adoption is accelerating across industries. A geotechnical engineering review (2025) shows publications on PIML grew exponentially after 2019, with hundreds of papers by 2024.
Adoption Drivers
Research Community: Over 2 million researchers use AlphaFold database across 190 countries
Commercial Interest: Major tech companies (Google, NVIDIA, Huawei, Baidu, ByteDance) are deploying PIML systems
Regulatory Acceptance: Physics-informed models' interpretability appeals to regulators in healthcare, energy, and aerospace
Market Implications
While specific PIML market data is limited, related AI adoption is soaring. According to Founders Forum Group (July 2025):
73% of organizations worldwide use or pilot AI in core functions
AI adoption in UK companies grew from fewer than 250 in 2014 to over 1,400 by 2024
Emerging Applications
Digital Twins: PIML enables real-time virtual replicas of physical systems (Medium, March 2025)
Autonomous Systems: Self-driving vehicles and robots benefit from physics-informed planning
Climate Change Mitigation: PIML models help design carbon capture systems and optimize renewable energy
Personalized Medicine: Patient-specific models using physiological PDEs
Materials Discovery: Accelerated design of new materials with desired properties
Academic Focus
Leading research institutions have established dedicated programs:
Brown University's PhILMs Center (Physics-Informed Learning Machines for Multiscale and Multiphysics Problems)
Multiple universities offering courses on scientific machine learning
NeurIPS conference series on "Machine Learning and the Physical Sciences" (2025)
Future Trajectory (2025-2030)
Several trends will shape PIML's evolution:
Foundation Models: Development of pre-trained PIML models that transfer across domains
Hybrid Intelligence: Combining human expertise, traditional simulation, and PIML
Uncertainty Quantification: Better methods for quantifying prediction confidence
AutoML for Physics: Automated tools for non-experts to build physics-informed models
Hardware Acceleration: Specialized chips for PDE computations and automatic differentiation
Regulatory Frameworks: Standards for deploying PIML in safety-critical applications
A 2024 review in AI notes: "PIML is expected to play a crucial role in enhancing maintenance strategies, system reliability, and overall operational efficiency in engineering systems."
Tools and Frameworks
Open-Source Libraries
DeepXDE: Python library for solving PDEs with PINNs, developed by Lu Lu and George Karniadakis
Supports multiple backend frameworks (TensorFlow, PyTorch, JAX)
Includes examples for dozens of PDE problems
Active development and community
NeuralPDE.jl: Julia package for physics-informed learning
Leverages Julia's performance for scientific computing
Integrated with DifferentialEquations.jl ecosystem
ADCME (Automatic Differentiation for Computational Mechanics): Julia-based framework
Designed for inverse problems in computational mechanics
GPU acceleration support
PyDEns: Python library for deep learning of PDE solutions
Focus on research and experimentation
SciANN: Python library built on TensorFlow/Keras
User-friendly API for PINNs
Good for beginners
Commercial Platforms
MATLAB Deep Learning Toolbox: Includes PINN capabilities as of recent versions
Integration with MATLAB's simulation ecosystem
Documentation and examples for engineers
NVIDIA Modulus: Platform for physics-ML models
Optimized for NVIDIA GPUs
Pre-built physics-informed components
Cloud Resources
Google Colab: Free GPU access for PIML experimentation
AlphaFold Server: Free protein structure prediction (Google DeepMind, 2024)
Frequently Asked Questions
Q1: Is PIML the same as scientific machine learning?
Yes, the terms are largely synonymous. Scientific Machine Learning (SciML) is the most common alternative name. Other variations include Process-Guided Machine Learning, Knowledge-Guided ML, and Theory-Trained Neural Networks.
Q2: Do I need to know physics to use PIML?
It depends. Using existing PIML frameworks requires moderate physics understanding. Building new PIML models for novel problems demands expertise in both the domain physics and machine learning.
Q3: Can PIML work with incomplete physics knowledge?
Yes, partially. PIML models can learn corrections to approximate physical models. Research shows hybrid approaches where a neural network learns the "residual physics" not captured by simplified equations.
Q4: How much data does PIML need?
Significantly less than pure ML—often 60-90% less. However, you still need some data. The exact amount depends on problem complexity and how accurately you know the physics.
Q5: Is PIML faster than traditional simulation?
For inference (making predictions), yes—typically 100-1000x faster. Training can be computationally intensive, but once trained, PIML models run in milliseconds.
Q6: Can PIML handle noisy data?
Yes. The physics constraints help filter noise. PIML models often outperform pure ML on noisy datasets because physics provides regularization.
Q7: What programming languages work for PIML?
Python dominates (via TensorFlow, PyTorch, JAX). Julia is growing (better for scientific computing). Some MATLAB support exists.
Q8: Are PINNs the only form of PIML?
No. PINNs are the most popular, but many alternatives exist: physics-informed feature engineering, Hamiltonian neural networks, physics-constrained architectures, symbolic regression with physics constraints, and more.
Q9: Can PIML solve inverse problems?
Absolutely—this is a major strength. PIML can infer unknown parameters (material properties, boundary conditions, source terms) from limited observations.
Q10: How does PIML compare to transfer learning?
They're complementary. Transfer learning reduces data needs by leveraging pre-trained models. PIML reduces data needs through physics constraints. Combining both (like transfer learning with PINNs) yields even better results.
Q11: What's the biggest limitation of current PIML?
Computational training cost and implementation complexity. Training PINNs can be tricky, and debugging physics-informed loss functions requires expertise.
Q12: Is PIML just a research topic or actually used in industry?
Both. Active research continues, but real industrial deployments exist in weather forecasting, drug discovery, building management, manufacturing, and aerospace.
Key Takeaways
PIML bridges two worlds: It combines centuries of scientific knowledge with modern AI's pattern-recognition power
Data efficiency is transformative: Reduce training data by 60-90% compared to pure machine learning
Physical consistency matters: In safety-critical applications, models must obey natural laws—PIML guarantees this
PINNs revolutionized the field: Introduced in 2017, they're now applied across dozens of domains
Real impact today: Weather forecasts run 1000x faster, drug discovery accelerates by years, building energy use optimizes
Challenges remain: Training complexity, computational costs, and scaling to very high dimensions
Growing rapidly: Publications, commercial deployments, and open-source tools are exploding
Not a silver bullet: Works best when physics is well-understood and some data exists
Future is hybrid: Combining PIML with traditional simulation and human expertise yields best results
Accessible now: Open-source frameworks and cloud resources let anyone start experimenting
Actionable Next Steps
Start Learning: Take Stanford's course "CS229ML" or MIT's "Scientific Machine Learning" (available online)
Experiment: Install DeepXDE and run the heat equation example (takes 30 minutes)
Read Key Papers:
Raissi et al. "Physics-informed neural networks" (J. Comp. Phys., 2019)
Karniadakis et al. "Physics-informed machine learning" (Nature Reviews Physics, 2021)
Join Communities:
PIML subreddit
Scientific ML Slack channels
NeurIPS Physical Sciences workshops
Identify Use Cases: Look for problems in your domain with:
Known governing equations
Limited data
Need for fast repeated predictions
Physical consistency requirements
Start Simple: Begin with 1D problems before tackling complex 3D systems
Leverage Transfer Learning: Use pre-trained models when available
Consider Consulting: For mission-critical applications, engage PIML experts initially
Glossary
Automatic Differentiation (AD): Technique for computing derivatives numerically without symbolic math or finite differences. Essential for PINNs.
Boundary Conditions: Constraints on solution at problem boundaries (e.g., temperature at wall surface).
Computational Fluid Dynamics (CFD): Traditional numerical simulation of fluid flow using discretized equations.
Conservation Laws: Physical principles stating certain quantities (mass, energy, momentum) remain constant in closed systems.
Domain Decomposition: Breaking large problems into smaller subproblems, solved separately then combined.
Finite Element Method (FEM): Traditional numerical technique discretizing equations on a mesh.
Forward Problem: Predicting system behavior from known parameters and conditions.
Hamiltonian: Function describing total energy in a physical system, used in Hamiltonian neural networks.
Initial Conditions: Starting values for time-dependent problems.
Inverse Problem: Inferring unknown parameters from observed behavior (harder than forward problems).
Loss Function: Objective function neural networks minimize during training.
Neural Operator: Neural network that learns mappings between function spaces (not just vectors).
Partial Differential Equation (PDE): Equation involving functions and their partial derivatives, describing many physical phenomena.
Residual: Degree to which a solution fails to satisfy an equation (should be zero for exact solutions).
Scientific Machine Learning (SciML): Synonym for physics-informed machine learning.
Surrogate Model: Fast approximation replacing expensive high-fidelity simulations.
Uncertainty Quantification (UQ): Estimating confidence and error bounds in predictions.
Sources & References
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. https://doi.org/10.1038/s42254-021-00314-5
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://doi.org/10.1016/j.jcp.2018.10.045
Faegh, M., Ghungrad, S., Oliveira, J. P., Rao, P., & Haghighi, A. (2025). A review on physics-informed machine learning for process-structure-property modeling in additive manufacturing. Journal of Manufacturing Processes, 133, 524-555. https://doi.org/10.1016/j.jmapro.2024.11.066
Ma, Z., Jiang, G., Hu, Y., & Chen, J. (2025). A review of physics-informed machine learning for building energy modeling. Applied Energy, 381, 125169. https://doi.org/10.1016/j.apenergy.2024.125169
Zhang, Q., Chen, Y., & Yang, Z. et al. (2025). When physics meets machine learning: a survey of physics-informed machine learning. Machine Learning for Computational Science and Engineering. https://doi.org/10.1007/s44379-025-00016-0
Cuomo, S., Di Cola, V. S., Giampaolo, F., et al. (2022). Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What's Next. Journal of Scientific Computing, 92, 88. https://doi.org/10.1007/s10915-022-01939-z
Kashinath, K., Mustafa, M., Albert, A., et al. (2021). Physics-informed machine learning: case studies for weather and climate modelling. Philosophical Transactions of the Royal Society A, 379(2194), 20200093. https://doi.org/10.1098/rsta.2020.0093
Toscano, J. D., Oommen, V., Varghese, A. J., et al. (2024). From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning. arXiv:2410.13228. https://arxiv.org/abs/2410.13228
Abramson, J., et al. (2024). Accurate structure prediction of biomolecular interactions with AlphaFold 3. Nature. https://doi.org/10.1038/s41586-024-07487-w
Jumper, J., Evans, R., Pritzel, A., et al. (2021). Highly accurate protein structure prediction with AlphaFold. Nature, 596, 583–589. https://doi.org/10.1038/s41586-021-03819-2
Sharma, P., Chung, W. T., Akoush, B., & Ihme, M. (2023). A review of physics-informed machine learning in fluid mechanics. Energies, 16(5), 2343. https://doi.org/10.3390/en16052343
Zhou, J., Zhou, Y., Wang, S., et al. (2024). Physics-informed machine learning: A comprehensive review on applications in anomaly detection and condition monitoring. Expert Systems with Applications, 251, 124015. https://doi.org/10.1016/j.eswa.2024.124015
Wang, L., Chan, Y.-C., Ahmed, F., Liu, Z., Zhu, P., & Chen, W. (2022). Deep learning of parametric partial differential equations from sparse and noisy data. Physics of Fluids, 33, 037132. https://doi.org/10.1063/5.0042868
Zhao, C., Zhang, F., Lou, W., Wang, X., & Yang, J. (2024). A comprehensive review of advances in physics-informed neural networks and their applications in complex fluid dynamics. Physics of Fluids, 36(10), 101301. https://doi.org/10.1063/5.0226562
Pacific Northwest National Laboratory. Physics-Informed Machine Learning. https://www.pnnl.gov/explainer-articles/physics-informed-machine-learning
MATLAB. (2024). What Are Physics-Informed Neural Networks (PINNs)? https://www.mathworks.com/discovery/physics-informed-neural-networks.html
Google DeepMind. (2024). AlphaFold - Revealing the structure of the protein universe. https://deepmind.google/science/alphafold/
Founders Forum Group. (2025). AI Statistics 2024-2025: Global Trends, Market Growth & Adoption Data. https://ff.co/ai-statistics-trends-global-market/
Zhou, K., et al. (2024). Data-Driven Weather Forecasting and Climate Modeling from the Perspective of Development. Atmosphere, 15(6), 689. https://doi.org/10.3390/atmos15060689
Wu, H., Chen, M., Cheng, H., Yang, T., Zeng, M., & Yang, M. (2025). Interpretable physics-informed machine learning approaches to accelerate electrocatalyst development. Journal of Materials Informatics, 2024:67. https://doi.org/10.48550/arXiv.2306.12059
$50
Product Title
Product Details goes here with the simple product description and more information can be seen by clicking the see more button. Product Details goes here with the simple product description and more information can be seen by clicking the see more button
$50
Product Title
Product Details goes here with the simple product description and more information can be seen by clicking the see more button. Product Details goes here with the simple product description and more information can be seen by clicking the see more button.
$50
Product Title
Product Details goes here with the simple product description and more information can be seen by clicking the see more button. Product Details goes here with the simple product description and more information can be seen by clicking the see more button.
Comments