Solving stochastic partial differential equations (SPDEs) using neural networks with a focus on Wiener chaos expansion provides a powerful approach to handling the inherent randomness in SPDEs. Here, I’ll explain the main concepts and how this framework is applied:

1. Background Concepts

SPDEs:

An SPDE typically has the form:

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

where:

• $u(t, x)$ is the unknown solution.
• $\mathcal{L}$ is a differential operator.
• $\sigma(\cdot)$ represents the diffusion coefficient.
• $\dot{W}(t, x)$ is a space-time white noise or more general stochastic process.

Wiener Chaos Expansion:

The Wiener chaos expansion is a series representation of random processes using orthogonal polynomials of Gaussian random variables (Hermite polynomials):

$u(t, x, \omega) = \sum_{k=0}^{\infty} u_k(t, x) H_k(\omega),$

where:

• $\omega$ represents randomness (e.g., outcomes in a probability space).
• $H_k(\omega)$ are Hermite polynomials, forming an orthogonal basis in the space of square-integrable functions.
• $u_k(t, x)$ are deterministic coefficients to be found.

2. Neural Networks in Solving SPDEs

The goal is to use neural networks to learn the deterministic components \( u_k(t, x) \) of the solution. These components can then be used to reconstruct the solution in the Wiener chaos framework.

Approach:

1. Representation:

   • The solution $u(t, x, \omega)$ of the SPDE is represented using the Wiener chaos expansion, where neural networks approximate the deterministic coefficients $u_k(t, x)$ .
   • $u_k(t, x)$ are trained neural networks denoted as $u_k^\theta(t, x)$ , with $\theta$ representing network parameters.

2. Training Objective:

   • A loss function is constructed to minimize the error between the left-hand side and right-hand side of the SPDE. The loss incorporates both deterministic and stochastic components.

3. Loss Function:

   • The loss is formulated as:

   $\mathcal{L}(\theta) = \mathbb{E} \left[ \left( \frac{\partial u_\theta(t, x)}{\partial t} - \mathcal{L}u_\theta(t, x) - \sigma(u_\theta(t, x)) \dot{W}(t, x) \right)^2 \right],$

   where $u_\theta(t, x)$ is the neural network's approximation.

   • The expected value $\mathbb{E}[\cdot]$ can be approximated using Monte Carlo or other numerical integration methods.

4. Wiener Chaos Expansion in Neural Networks:

   • Each coefficient $u_k(t, x)$ can be learned using a separate neural network or collectively as output nodes of a single network structure.
   • The network should be trained to minimize the loss such that the learned coefficients $u_k(t, x)$ align with the series expansion of the solution.