解决科学神经求解器的病态问题，让计算更准更快。
Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equations
- 用平移高斯编码过滤激活矩阵，提升数值稳定性
- 稳态对流-扩散方程的Peclet数范围扩大超100倍，误差降6个数量级
- 适合需要高精度与快速收敛的物理建模场景
神经微分方程求解器的精度下降常非因表达能力不足，而是由病态条件导致的优化困难，尤其在多保真度和刚性问题中更为显著。本文研究了物理信息极值学习机（PIELM）中的此类问题，发现控制方程中的渐近成分会导致激活矩阵严重病态，极大限制收敛性。为此提出平移高斯编码（Shifted Gaussian Encoding），一种简单有效的激活过滤机制，在保持凸性的前提下提升矩阵秩与表达能力。该方法使稳态对流-扩散方程的可解佩克莱特数（Peclet number）范围扩大超过两个数量级，多频函数学习误差降低达六个数量级，并能更准确、快速地拟合高保真图像向量，优于参数超过百万的深度网络。本工作表明，科学神经求解器的关键瓶颈在于条件数而非模型深度，简单的架构改进即可带来显著性能提升。
原文摘要 · Abstract (English)
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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