标题： 快速、凸性和条件网络用于多保真度向量和刚性一元微分方程 显示英文标题

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

摘要： 神经PDE求解器的准确性往往不是因为表达能力有限，而是由于病态条件导致的优化不佳，尤其是在多保真度和刚性问题中。我们研究了这一问题，在物理信息极端学习机（PIELMs）中，这是一种凸变体的神经PDE求解器，我们展示了控制方程中的渐近成分会产生高度病态的激活矩阵，严重限制收敛。我们引入了移位高斯编码，这是一种简单而有效的激活过滤步骤，可以在保持凸性的前提下增加矩阵秩和表达能力。我们的方法将稳态对流-扩散方程中的佩克莱特数的可解范围扩展了两个数量级，在多频函数学习上实现了误差降低六个数量级，并且比具有超过一百万参数的深度网络更准确、更快地拟合高保真图像向量。这项工作强调了条件性而不是深度通常是科学神经求解器的瓶颈，并且简单的架构变化可以释放显著的收益。 显示英文摘要

摘要： 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.