High Precision Differentiation Techniques for Data-Driven Solution of Nonlinear PDEs by Physics-Informed Neural Networks

TL;DR

This paper introduces new differentiation techniques for time-dependent PDEs that enable accurate higher order derivatives, improving data-driven solutions with Physics-Informed Neural Networks for real-world equations.

Contribution

It proposes novel differentiation methods for PDE solutions that enhance the accuracy of derivatives used in physics-informed neural network models.

Findings

Accurate higher order derivatives generated at multiple spatial points.

Application demonstrated on Burgers', Allen-Cahn, and Schrödinger equations.

Improved data-driven PDE solutions using the new techniques.

Abstract

Time-dependent Partial Differential Equations with given initial conditions are considered in this paper. New differentiation techniques of the unknown solution with respect to time variable are proposed. It is shown that the proposed techniques allow to generate accurate higher order derivatives simultaneously for a set of spatial points. The calculated derivatives can then be used for data-driven solution in different ways. An application for Physics Informed Neural Networks by the well-known DeepXDE software solution in Python under Tensorflow background framework has been presented for three real-life PDEs: Burgers', Allen-Cahn and Schrodinger equations.