High Precision Differentiation Techniques for Data-Driven Solution of Nonlinear PDEs by Physics-Informed Neural Networks
This work addresses the challenge of improving accuracy in solving PDEs for computational science and engineering, though it appears incremental as it builds on existing PINN methods.
The paper tackles the problem of solving time-dependent nonlinear PDEs by proposing new differentiation techniques for generating accurate higher-order time derivatives, which are applied using Physics-Informed Neural Networks (PINNs) via DeepXDE software to achieve data-driven solutions for Burgers', Allen-Cahn, and Schrodinger equations.
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.