Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equations

• URL: http://arxiv.org/abs/2508.05921v1
• Date: Fri, 08 Aug 2025 00:51:38 GMT
• Title: Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equations
• Authors: Siddharth Rout,
• Abstract summary: Accuracy in neural PDE solvers often breaks down due to poor optimisation caused by ill-conditioning.<n>We show that components in governing equations can produce highly ill-conditioned activation vectors.<n>We introduce Shifted Gaussian, a simple yet effective activation filtering step that increases matrix rank and expressivity while preserving convexity.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Accuracy in neural PDE solvers often breaks down not because of limited expressivity, but due to poor optimisation caused by ill-conditioning, especially in multi-fidelity and stiff problems. We study this issue in Physics-Informed Extreme Learning Machines (PIELMs), a convex variant of neural PDE solvers, and show that asymptotic components in governing equations can produce highly ill-conditioned activation matrices, severely limiting convergence. We introduce Shifted Gaussian Encoding, a simple yet effective activation filtering step that increases matrix rank and expressivity while preserving convexity. Our method extends the solvable range of Peclet numbers in steady advection-diffusion equations by over two orders of magnitude, achieves up to six orders lower error on multi-frequency function learning, and fits high-fidelity image vectors more accurately and faster than deep networks with over a million parameters. This work highlights that conditioning, not depth, is often the bottleneck in scientific neural solvers and that simple architectural changes can unlock substantial gains.

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