
Quantifying total uncertainty in physicsinformed neural networks for solving forward and inverse stochastic problems
Physicsinformed neural networks (PINNs) have recently emerged as an alt...
09/21/2018 ∙ by Dongkun Zhang, et al. ∙ 0 ∙ shareread it

Physics Informed Deep Learning (Part II): Datadriven Discovery of Nonlinear Partial Differential Equations
We introduce physics informed neural networks  neural networks that ar...
11/28/2017 ∙ by Maziar Raissi, et al. ∙ 0 ∙ shareread it

DeepXDE: A deep learning library for solving differential equations
Deep learning has achieved remarkable success in diverse applications; h...
07/10/2019 ∙ by Lu Lu, et al. ∙ 33 ∙ shareread it

Solving Parameter Estimation Problems with Discrete Adjoint Exponential Integrators
The solution of inverse problems in a variational setting finds best est...
04/09/2017 ∙ by Ulrich Roemer, et al. ∙ 0 ∙ shareread it

Dynamically orthogonal tensor methods for highdimensional nonlinear PDEs
We develop new dynamically orthogonal tensor methods to approximate mult...
07/12/2019 ∙ by Alec Dektor, et al. ∙ 0 ∙ shareread it

Optimal Estimation of Dynamically Evolving Diffusivities
The augmented, iterated Kalman smoother is applied to system identificat...
03/11/2018 ∙ by Kurt S. Riedel, et al. ∙ 0 ∙ shareread it

From Deep to PhysicsInformed Learning of Turbulence: Diagnostics
We describe physical tests validating progress made toward acceleration ...
10/16/2018 ∙ by Ryan King, et al. ∙ 0 ∙ shareread it
Learning in Modal Space: Solving TimeDependent Stochastic PDEs Using PhysicsInformed Neural Networks
One of the open problems in scientific computing is the longtime 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 biorthogonal (BO) methods for representing stochastic processes. Specifically, we propose two new PhysicsInformed Neural Networks (PINNs) for solving timedependent SPDEs, namely the NNDO/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 NNDO/BO methods can be used to solve timedependent 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) Longtime 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 NNDO/BO methods in handling inverse and mixed type problems. Taken together, these simulation results demonstrate that the NNDO/BO methods can be employed to effectively quantify uncertainty propagation in a wide range of physical problems.
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