A Structure-Preserving Framework for Solving Parabolic Partial Differential Equations with Neural Networks
This work addresses the issue of physical inconsistency in neural network PDE solvers for scientific and engineering applications, representing an incremental improvement by enhancing existing methods.
The authors tackled the problem of neural network solvers for parabolic PDEs lacking physical consistency, which can lead to nonphysical or unstable solutions, and proposed the Sidecar framework to enhance physical consistency, resulting in significant improvements in accuracy and structure preservation.
Solving partial differential equations (PDEs) with neural networks (NNs) has shown great potential in various scientific and engineering fields. However, most existing NN solvers mainly focus on satisfying the given PDE formulas in the strong or weak sense, without explicitly considering some intrinsic physical properties, such as mass and momentum conservation, or energy dissipation. This limitation may result in nonphysical or unstable numerical solutions, particularly in long-term simulations. To address this issue, we propose ``Sidecar'', a novel framework that enhances the physical consistency of existing NN solvers for solving parabolic PDEs. Inspired by the time-dependent spectral renormalization approach, our Sidecar framework introduces a small network as a copilot, guiding the primary function-learning NN solver to respect the structure-preserving properties. Our framework is highly flexible, allowing the preservation of various physical quantities for different PDEs to be incorporated into a wide range of NN solvers. Experimental results on some benchmark problems demonstrate significant improvements brought by the proposed framework to both accuracy and structure preservation of existing NN solvers.