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

• URL: http://arxiv.org/abs/2507.20929v1
• Date: Mon, 28 Jul 2025 15:41:51 GMT
• Title: Breaking the Precision Ceiling in Physics-Informed Neural Networks: A Hybrid Fourier-Neural Architecture for Ultra-High Accuracy
• Abstract summary: Physics-informed neural networks (PINNs) have plateaued at errors of $10-3$-$10-4$ for fourth-order partial differential equations.<n>We achieve unprecedented L2 error of $1.94 times 10-7$-a 17-fold improvement over standard PINNs and (15-500times) better than traditional numerical methods.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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