Geometry-adaptive physics-informed neural solver for parametric partial differential equations

Anonymous

Keywords: Physics-informed neural networks (PINNs), geometry-adaptive physics-informed neural solver, generic solvers for real-world simulations, rethinking PINNs in geometry-adaptive space, Fourier neural operators

TL;DR: A generic solver, geometry-adaptive physics-informed neural solver with resolution-invariant, geometry-adaptive properties, is proposed to solve Partial Differential Equations (PDEs).

Abstract: Partial differential equations (PDEs) are notoriously difficult to solve, and traditional numerical approximation schemes have high computational costs. Recently, hybrid neural-numerical solvers have been developed to ride on the modern trend of fully end-to-end learning systems. Typical approaches can be divided into physics-informed neural networks (PINNs), which approximately satisfy a given set of PDEs by constraining neural networks, and neural solvers, which learn solutions by training data. However, these solvers cannot fulfil the properties of the generic solver for real-world simulations, which should be fast, stable, accurate, no training data, multi-scale and multi-physics, resolution invariance, geometric adaptation. In this work, we build a geometry-adaptive physics-informed neural solver (GeoPINS) that satisfies these properties, combining the advantages of no training data in PINNs, as well as fast, accurate and resolution-invariant architectures offered by Fourier neural operators (FNO). In particular, to adapt to complex and irregular geometries that exist in the real world, we reformulate PINNs in geometry-adaptive space by taking full advantage of coordinate transformations and the efficiency of numerical methods (including pseudo-spectral and high-order finite difference methods) in solving the spatial gradient. In order to overcome observed issues of conventional PINNs when solving challenging PDEs (such as high-frequency, multi-physics), we rewrite FNO as an efficient visual mixer to construct spatial-temporal representations and validate our approach on many popular PDEs on both regular and irregular domains, demonstrating fast, stable, and accurate performance, as well as resolution-invariant, geometry-adaptive properties for real-world simulations. In addition, GeoPINS achieves superior accuracy compared to previous solvers at different zero-shot super-resolution settings.