Breaking the Precision Ceiling in Physics-Informed Neural Networks: A Hybrid Fourier-Neural Architecture for Ultra-High Accuracy

TL;DR

This paper introduces a hybrid Fourier-neural network architecture that significantly improves the accuracy of physics-informed neural networks for high-order PDEs, surpassing previous error limits and rivaling traditional numerical methods.

Contribution

The authors develop a novel hybrid Fourier-neural architecture with a systematic harmonic optimization, enabling ultra-high accuracy in PINNs for the Euler-Bernoulli beam equation, overcoming the previous precision ceiling.

Findings

Achieved an L2 error of 1.94e-7, a 17-fold improvement over standard PINNs.

Optimal performance with exactly 10 harmonics, with accuracy dropping sharply beyond this.

Stable convergence achieved with a two-phase optimization and adaptive weight balancing.

Abstract

Physics-informed neural networks (PINNs) have plateaued at errors of $10^{-3}$-$10^{-4}$ for fourth-order partial differential equations, creating a perceived precision ceiling that limits their adoption in engineering applications. We break through this barrier with a hybrid Fourier-neural architecture for the Euler-Bernoulli beam equation, achieving unprecedented L2 error of $1.94 \times 10^{-7}$-a 17-fold improvement over standard PINNs and \(15-500\times\) better than traditional numerical methods. Our approach synergistically combines a truncated Fourier series capturing dominant modal behavior with a deep neural network providing adaptive residual corrections. A systematic harmonic optimization study revealed a counter-intuitive discovery: exactly 10 harmonics yield optimal performance, with accuracy catastrophically degrading from $10^{-7}$ to $10^{-1}$ beyond this threshold. The two-phase optimization strategy (Adam followed by L-BFGS) and adaptive weight balancing enable stable ultra-precision convergence. GPU-accelerated implementation achieves sub-30-minute training despite fourth-order derivative complexity. By addressing 12 critical gaps in existing approaches-from architectural rigidity to optimization landscapes-this work demonstrates that ultra-precision is achievable through proper design, opening new paradigms for scientific computing where machine learning can match or exceed traditional numerical methods.