发表机构
Indian Institute of Technology Jodhpur（印度理工学院焦特布尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出在PINN第一层引入输入依赖的局部化函数，无需域划分即可提升长域和高阶问题求解精度，筛选出逆二次族在三个基准上均优于基线。
AI 中文摘要
物理信息神经网络（PINNs）在计算域上使用一个共享表示，这在长域和高阶算子问题上可能难以优化。我们研究了一种极简替代方案：将原本不变的稠密PINN的第一层隐藏激活乘以依赖于输入的局部化函数，从而在不划分域或添加界面损失的情况下，赋予第一层单元感受野。我们在非线性谐振子（HO）、长空间区间上的热方程以及一个构造的四维（4D）四阶问题上，对13族局部化函数（每族最多三种参数化）进行了筛选，全程使用十对配对种子。三种配置在匹配预算下大幅降低了求解误差：（i）在$2\pi$ HO域上，固定高斯局部化函数在3k轮次时将平均解RMSE从$4.8369\times10^{-1}$降至$8.83\times10^{-3}$。（ii）在$8\pi$热域上，具有可学习中心和宽度的逆二次族在10k轮次时将其从$2.896\times10^{-1}$降至$3.06\times10^{-2}$。（iii）在$4\pi$ 4D域上，固定凸起局部化函数在10k轮次时将其从$1.75947\times10^{1}$降至$2.260\times10^{-1}$。在这三项比较中，每一对配对种子都有所改善。筛选还表明该机制并非免费午餐：在HO上，13族中只有2族优于基线，其余11族中有10族差9至23倍；在4D上，四族产生非有限值，五族比基线差三个数量级以上。逆二次族是唯一在所有三个方程上都优于基线的。总体而言，这些结果表明，第一层局部化可以为基线PINN在长域和高阶问题上带来可衡量的改进。
英文摘要
Physics-informed neural networks (PINNs) use one shared representation over the computational domain, which can become difficult to optimize on long domains and for high-order operators. We study a minimal alternative: multiply the first hidden activation of an otherwise unchanged dense PINN by input-dependent localization functions, giving first-layer units receptive fields without partitioning the domain or adding interface losses. We screen 13 families of localization functions, in up to three parameterizations each, on a nonlinear harmonic oscillator (HO), a heat equation on a long spatial interval, and a manufactured four-dimensional (4D) fourth-order problem, with ten paired seeds throughout. Three configurations give large reductions in solution error at matched budgets: (i) Fixed Gaussian localization functions on the $2π$ HO domain cut mean solution RMSE from $4.8369\times10^{-1}$ to $8.83\times10^{-3}$ at 3k epochs. (ii) The inverse-quadratic family with learnable centers and widths cuts it from $2.896\times10^{-1}$ to $3.06\times10^{-2}$ on the $8π$ heat domain at 10k epochs. (iii) Fixed bump localization functions cut it from $1.75947\times10^{1}$ to $2.260\times10^{-1}$ on the $4π$ 4D domain at 10k epochs. Every paired seed improves in these three comparisons. The screen also shows that the mechanism is not a free win: on HO only 2 of 13 families beat the baseline, and 10 of the remaining 11 are 9 to 23 times worse; on 4D four families are non-finite and five are more than three orders of magnitude worse than the baseline. The inverse-quadratic family is the only one that beats the baseline on all three equations. Overall, these results show that first-layer localization can provide measurable improvements to baseline PINNs on long-domain and high-order problems.
