发表机构
Portland State University; University of South Carolina（波特兰州立大学; 南卡罗来纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对参数单调非线性偏微分方程，构造了可计算且全局可靠高效的误差估计器，兼作神经网络损失函数，并给出显式常数的双侧界。
AI 中文摘要
我们为具有单调性的一类参数非线性偏微分方程构造了可计算的误差估计器，这些估计器同时可作为神经网络的损失函数，并证明了它们是全局可靠且高效的。对于每一个试验函数，此类损失函数的值都受到自然试验范数下平方误差的上下界限制，而不仅仅是针对接近精确解的那些函数；这一全局性质依赖于单调性。该构造基于将非线性算子分解为线性部分和强单调闭包，并在离散对偶范数中度量线性残差。由于当试验范数强于$L_2$范数时，闭包所贡献的对偶范数没有闭式解，因此估计器围绕其可计算的替代量构建，该替代量仅需满足配对界和Lipschitz界。主定理随后给出具有显式常数的双侧界，并涵盖两个实例。第一个实例是在协调试验空间上的一阶系统最小二乘估计器：其双侧界在整个试验空间上成立，因此适用于任意逼近，包括那些不在离散有限元空间中的逼近。第二个实例是在每个参数值的有限元函数试验空间上的不连续Petrov-Galerkin估计器，使用破碎的测试空间构建，无需协调性，对偶范数通过独立的单元局部问题计算。所有假设均针对一类非线性通量模型得到验证，常数以参数范围显式追踪。由于对任意输入均可计算，这两个估计器可作为参数到解映射的神经网络逼近的变分正确损失函数。
英文摘要
We construct computable error estimators, which double as loss functions for neural networks, for a class of parametric nonlinear partial differential equations with a monotonicity property, and prove that they are globally reliable and efficient. The value of such a loss function is bounded above and below by the squared error in the natural trial norm, for every trial function, not merely for those near the exact solution; this global property rests on monotonicity. The construction rests on splitting the nonlinear operator into a linear part and a strongly monotone closure, and on measuring the linear residual in a discrete dual norm. Since the closure contributes a dual norm that admits no closed form when the trial norm is stronger than an $L_2$ norm, the estimator is built around a computable surrogate for it, required only to satisfy a pairing bound and a Lipschitz bound. The main theorem then yields two-sided bounds with explicit constants and covers two instances. A first-order system least-squares estimator on conforming trial spaces is the first instance studied: its two-sided bound holds on the whole trial space and therefore applies to arbitrary approximations, including those not in a discrete finite element space. The second instance is a discontinuous Petrov-Galerkin estimator on trial spaces of finite element functions for each parameter value, built with broken test spaces, for which no conformity is required and the dual norm is computed by independent element-local problems. All assumptions are verified for a model class of nonlinear fluxes, with the constants tracked explicitly in terms of the parameter range. Being computable for an arbitrary input, both estimators serve as variationally correct loss functions for neural network approximations of parameter-to-solution maps.
Comments30 pages, 2 figures, 3 tables
