Physics-Informed Machine Learning

Extend deep learning workflows in areas of physics-informed machine learning (PIML) and physics-informed neural networks (PINNs)

Use Deep Learning Toolbox™ for physics-informed machine learning (PIML) and physics-informed neural networks (PINNs).

Physics-informed machine learning (PIML) and physics-informed neural networks refer to machine learning and deep learning concepts where you can integrate laws and principles of physical systems into your machine learning models. Integrating these concepts can improve accuracy and robustness in these models and can help ensure that the model predictions also follow such laws and principles. For example, you can train a neural network that models heat transfer using a loss function that incorporates laws of thermodynamics.

Functions

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Neural Network Layers

`neuralODELayer`	Neural ODE layer (Since R2023b)
`complexToRealLayer`	Complex-to-real layer (Since R2024b)
`realToComplexLayer`	Real-to-complex layer (Since R2024b)
`complexReluLayer`	Complex rectified linear unit (ReLU) layer (Since R2025a)

Automatic Differentiation

`dlarray`	Deep learning array for customization
`dlgradient`	Compute gradients for custom training loops using automatic differentiation
`dljacobian`	Jacobian matrix deep learning operation (Since R2024b)
`dldivergence`	Divergence of deep learning data (Since R2024b)
`dllaplacian`	Laplacian of deep learning data (Since R2024b)
`dlfeval`	Evaluate deep learning model for custom training loops
`dlode45`	Deep learning solution of nonstiff ordinary differential equation (ODE) (Since R2021b)

Topics

Solve PDE Using Fourier Neural Operator
This example shows how to train a Fourier neural operator (FNO) neural network that outputs the solution of a partial differential equation (PDE).
Solve PDE Using Physics-Informed Neural Network
This example shows how to train a physics-informed neural network (PINN) to predict the solutions of an partial differential equation (PDE).
Solve ODE Using Physics-Informed Neural Network
This example shows how to train a physics-informed neural network (PINN) to predict the solutions of an ordinary differential equation (ODE).
Train Latent ODE Network with Irregularly Sampled Time-Series Data
This example shows how to train a latent ordinary differential equation (ODE) autoencoder with time-series data that is sampled at irregular time intervals.
Dynamical System Modeling Using Neural ODE
This example shows how to train a neural network with neural ordinary differential equations (ODEs) to learn the dynamics of a physical system.
Solve Inverse Problem for PDE Using Physics-Informed Neural Network
This example shows how to solve an inverse problem using a physics-informed neural network (PINN).
Solve Poisson Equation on Unit Disk Using Physics-Informed Neural Networks (Partial Differential Equation Toolbox)
Solve a Poisson's equation with Dirichlet boundary conditions using a physics-informed neural network (PINN).

Related Information

Scientific and Physics Informed Machine Learning in MATLAB (GitHub)

Featured Examples

Solve Heat Equation Using Graph Neural Network

Solve a heat equation with using a graph neural network (GNN).

(Partial Differential Equation Toolbox)

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