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====== Physics-Informed Neural Networks (PINNs) ======

**Physics-Informed Neural Networks (PINNs)** are a class of deep learning models that embed known physical laws — expressed as partial differential equations (PDEs) or ordinary differential equations (ODEs) — directly into the neural network training process as soft constraints in the loss function.((M. Raissi, P. Perdikaris, and G.E. Karniadakis, "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, 2019. [[https://www.sciencedirect.com/science/article/abs/pii/S0021999118307125|ScienceDirect]])) This approach bridges data-driven machine learning with first-principles physics, enabling solutions that respect conservation laws, boundary conditions, and governing equations even with sparse or noisy data.

===== Core Concept =====

A PINN approximates the solution to a differential equation by training a neural network whose loss function includes three components:

  * **Data loss (L_D)** — error between network predictions and available observational data
  * **Physics loss (L_F)** — residual of the governing PDE/ODE evaluated at collocation points, computed via automatic differentiation
  * **Boundary/initial condition loss (L_B)** — error at domain boundaries or initial time steps

The composite loss is:

  L = L_D + L_F + L_B

The network learns to minimize all three simultaneously, producing solutions that are consistent with both observed data and known physics.((Steve Brunton, "Physics Informed Neural Networks (PINNs)," University of Washington, 2024. [[https://www.youtube.com/watch?v=-zrY7P2dVC4|YouTube]]))

===== Training Process =====

  - Define a neural network with inputs (spatial/temporal coordinates, parameters) and outputs (physical field quantities)
  - Formulate the composite loss from data, PDE residuals, and boundary conditions
  - Sample collocation points throughout the domain
  - Compute PDE residuals using automatic differentiation (backpropagation computes the partial derivatives needed)
  - Optimize using Adam followed by L-BFGS for fine-tuning
  - Enforce boundary conditions either softly (as penalty terms) or via hard constraints built into the network architecture
  - Validate against known solutions or experimental data

===== Advantages =====

  * **Mesh-free** — no computational mesh is required, unlike finite element or finite volume methods
  * **Data efficiency** — physics constraints regularize the network, enabling learning from sparse data
  * **Inverse problems** — PINNs can infer unknown parameters (e.g., material properties, diffusion coefficients) from observed data
  * **High-dimensional problems** — neural networks handle high-dimensional PDEs that cause the "curse of dimensionality" in traditional solvers
  * **Continuous solutions** — outputs are smooth, differentiable functions rather than discrete grid values

===== Limitations =====

  * **Training difficulty** — balancing the multiple loss terms can be challenging; the physics loss and data loss may compete
  * **Spectral bias** — neural networks tend to learn low-frequency components first, struggling with high-frequency or sharp features like shock waves((P. Kumar and R. Ranjan, "A robust data-free physics-informed neural network for compressible flows with shocks," Computers & Fluids, 308:106975, March 2026. [[https://www.sciencedirect.com/science/article/abs/pii/S0045793026000174|ScienceDirect]]))
  * **Scalability** — for very large domains or long time integrations, training can be computationally expensive
  * **Accuracy vs. traditional solvers** — for well-posed forward problems with clean data, classical numerical methods remain more accurate and efficient

===== Applications =====

  * **Fluid dynamics** — modeling incompressible and compressible flows, turbulence, and aerodynamics
  * **Materials science** — predicting stress, strain, and failure in complex materials
  * **Weather and climate** — surrogate models for atmospheric dynamics
  * **Astrophysics** — modeling self-gravity in gas dynamics, gravitational waves, and cosmological simulations((M. Cieslar, "Physics-Informed Neural Networks (PINNs)," Astronomical Observatory, University of Warsaw, October 2025. [[https://www.astrouw.edu.pl/~rpoleski/sjc_files/slides/slides_2025_10_15_PINN.pdf|OAUW]]))
  * **Biomedical engineering** — hemodynamics, drug delivery modeling, and cardiac mechanics
  * **Energy systems** — battery modeling, heat transfer, and power grid simulation

===== Key Frameworks =====

  * **DeepXDE** — a Python library for PINNs supporting various PDE types, built on TensorFlow/PyTorch/JAX((L. Lu et al., "DeepXDE: A Deep Learning Library for Solving Differential Equations," SIAM Review, 2021.))
  * **NVIDIA Modulus** — an industrial-scale framework for physics-ML, providing GPU-accelerated PINN training and deployment for engineering applications((NVIDIA, "Modulus: A Framework for Physics-ML." [[https://developer.nvidia.com/modulus|developer.nvidia.com]]))
  * **PINA** — Physics-Informed Neural networks for Advanced modeling, supporting multiple PDE formulations((D. Coscia et al., "PINA: Physics-Informed Neural networks for Advanced modeling," JOSS, July 2023. [[https://joss.theoj.org/papers/10.21105/joss.05352|JOSS]]))
  * **SciANN** — Keras-based PINN library for scientific computing
  * **PyDEns** — a framework for solving PDEs with deep learning

===== Key Researchers =====

  * **George Em Karniadakis** (Brown University) — co-author of the foundational 2019 PINNs paper
  * **Maziar Raissi** — lead author of the original PINNs paper
  * **Paris Perdikaris** (University of Pennsylvania) — co-author, active in extending PINNs to multi-fidelity and operator learning
  * **Lu Lu** (Yale) — creator of DeepXDE

===== See Also =====

  * [[neural_operator|Neural Operators]]
  * [[scientific_machine_learning|Scientific Machine Learning]]
  * [[deep_learning|Deep Learning]]

===== References =====
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