Breaking the Precision Ceiling in Physics-Informed Neural Networks: A Hybrid Fourier-Neural Architecture for Ultra-High Accuracy
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Abstract (arXiv)
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 × 10-7-a 17-fold improvement over standard PINNs and (15-500×) 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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References · 20
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8 more
- 2021Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential trainingcited by 26
- 2023Physics-informed neural networks for solving forward and inverse problems in complex beam systemscited by 24
- 2023Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networkscited by 23
- 2024Dual Cone Gradient Descent for Training Physics-Informed Neural Networkscited by 21
- 2021A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticitycited by 12
- 2024Artificial intelligence for partial differential equations in computational mechanics: A reviewcited by 9
- 2023Transfer learning for improved generalizability in causal physics-informed neural networks for beam simulationscited by 8
- 2022High Precision Differentiation Techniques for Data-Driven Solution of Nonlinear PDEs by Physics-Informed Neural Networkscited by 2
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