Physics-informed neural networks to the solution of SPDEs

Physics-Informed Neural Networks (PINNs) have emerged as an innovative approach for solving partial differential equations (PDEs) and stochastic partial differential equations (SPDEs). By incorporating the governing physical laws directly into the training of neural networks, PINNs combine data-driven modeling with prior knowledge of the system, making them a powerful tool for solving SPDEs where traditional numerical methods may be computationally intensive or impractical.

1. What are Physics-Informed Neural Networks (PINNs)?

Concept: PINNs are neural networks trained not just with data but also guided by physical laws represented by PDEs or SPDEs. They minimize a composite loss function that includes terms for fitting data and satisfying differential equations.
Core Idea: The neural network $ u_\theta(t, x) $ is designed to approximate the solution of a PDE or SPDE, where $ \theta $ represents the network parameters. The network is trained such that its output respects the known physical dynamics.

2. Formulation for SPDEs

SPDEs often have the general form:

$\frac{\partial u(t, x)}{\partial t} = \mathcal{L}u(t, x) + \sigma(u(t, x)) \dot{W}(t, x),$

where $\mathcal{L}$ is a differential operator and $\dot{W}(t, x)$ represents a stochastic noise term.

In the context of PINNs, the solution $u(t, x)$ is approximated by a neural network $u_\theta(t, x)$ . The training process involves:

Deterministic Part: Ensuring that $u_\theta(t, x)$ approximately satisfies $\mathcal{L}u_\theta(t, x)$ in the PDE sense.
Stochastic Part: Incorporating noise effects by adding a stochastic term that matches the statistical properties of $\dot{W}(t, x)$ .

3. Training the Neural Network

The PINN framework defines a composite loss function, typically consisting of:

Residual Loss: Measures how well the neural network satisfies the SPDE:

$L {PDE}=\sum_i\left|\frac{\partial u\theta\left(t_i, x_i\right)}{\partial t}- L u_\theta\left(t_i, x_i\right)-\sigma\left(u_\theta\left(t_i, x_i\right)\right) \dot{W}\left(t_i, x_i\right)\right|^2$

where $(t_i, x_i)$ are sampled points from the domain.

Boundary and Initial Conditions Loss: Ensures the solution respects any given initial or boundary conditions.
Data-Driven Loss (Optional): Adds a term to fit observational data when available.

The total loss function $\mathcal{L}{\text{total}}$ *is:

$\mathcal{L}{\text{total}} = \lambda_{\text{PDE}} \mathcal{L}{\text{PDE}} + \lambda{\text{BC}} \mathcal{L}{\text{BC}} + \lambda{\text{data}} \mathcal{L}{\text{data}},$

where $\lambda{\text{PDE}}, \lambda_{\text{BC}},$* and $\lambda_{\text{data}}$ are weights balancing the contributions of each term.

4. Handling Stochastic Terms

For SPDEs, special considerations are needed:

Noise Approximation: White noise $\dot{W}(t, x)$ can be approximated using samples from a random process with known properties.
Monte Carlo Methods: Stochastic integrals appearing in the SPDE are often approximated using Monte Carlo simulations during the training phase.
Stochastic Loss Term: If the SPDE has statistical properties that can be characterized (e.g., mean and variance of the solution), these can be added to the loss function to ensure that the learned solution respects the statistical distribution.