A Fourier-neural PINN achieves 1.94e-7 L2 error on a beam equation, but the 'ultra-precision' is probably due to the solution being exactly representable by the chosen Fourier modes.
High Precision Differentiation Techniques for Data-Driven Solution of Nonlinear PDEs by Physics-Informed Neural Networks
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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.
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Breaking the Precision Ceiling in Physics-Informed Neural Networks: A Hybrid Fourier-Neural Architecture for Ultra-High Accuracy
A Fourier-neural PINN achieves 1.94e-7 L2 error on a beam equation, but the 'ultra-precision' is probably due to the solution being exactly representable by the chosen Fourier modes.