Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks

One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs). We address this problem by taking advantage of recent advances in scientific machine learning and the dynamically orthogonal (DO) and bi-orthogonal (BO) methods for representing stochastic processes. Specifically, we propose two new Physics-Informed Neural Networks (PINNs) for solving time-dependent SPDEs, namely the NN-DO/BO methods, which incorporate the DO/BO constraints into the loss function with an implicit form instead of generating explicit expressions for the temporal derivatives of the DO/BO modes. Hence, the proposed methods overcome some of the drawbacks of the original DO/BO methods: we do not need the assumption that the covariance matrix of the random coefficients is invertible as in the original DO method, and we can remove the assumption of no eigenvalue crossing as in the original BO method. Moreover, the NN-DO/BO methods can be used to solve time-dependent stochastic inverse problems with the same formulation and computational complexity as for forward problems. We demonstrate the capability of the proposed methods via several numerical examples: (1) A linear stochastic advection equation with deterministic initial condition where the original DO/BO method would fail; (2) Long-time integration of the stochastic Burgers' equation with many eigenvalue crossings during the whole time evolution where the original BO method fails. (3) Nonlinear reaction diffusion equation: we consider both the forward and the inverse problem, including noisy initial data, to investigate the flexibility of the NN-DO/BO methods in handling inverse and mixed type problems. Taken together, these simulation results demonstrate that the NN-DO/BO methods can be employed to effectively quantify uncertainty propagation in a wide range of physical problems.