This paper is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all PDE types for which a stable variational formulation (SVF) can be found. A central constituent is the notion of variationally correct residual loss function meaning that its value is always uniformly proportional to the squared solution error in the norm determined by the SVF, hence facilitating rigorous a posteriori accuracy control. It is based on a single variational problem, associated with the family of parameter dependent fiber problems, employing the notion of direct integrals of Hilbert spaces. Since in its original form the loss function is given as a dual test norm of the residual a central objective is to develop equivalent computable expressions. A first critical role is played by hybrid hypothesis classes, whose elements are piecewise polynomial in (low-dimensional) spatio-temporal variables with parameter-dependent coefficients that can be represented, e.g. by neural networks. Second, working with first order SVFs, we distinguish two scenarios: (i) the test space can be chosen as an $L_2$-space (e.g. for elliptic or parabolic problems) so that residuals live in $L_2$ and can be evaluated directly; (ii) when trial and test spaces for the fiber problems (e.g. for transport equations) depend on the parameters, we use ultraweak formulations. In combination with Discontinuous Petrov Galerkin concepts the hybrid format is then instrumental to arrive at variationally correct computable residual loss functions. Our findings are illustrated by numerical experiments representing (i) and (ii), namely elliptic boundary value problems with piecewise constant diffusion coefficients and pure transport equations with parameter dependent convection field.

翻译：本文研究学习依赖于潜在大量参数的偏微分方程（PDE）系统的参数-解映射，涵盖所有可找到稳定变分形式（SVF）的PDE类型。其核心构成是变分正确残差损失函数的概念，该函数值始终与SVF确定的范数中解误差的平方保持均匀比例，从而支持严格的后验精度控制。该方法基于单个变分问题，该问题与参数依赖的纤维问题族相关联，并运用希尔伯特空间直积分的概念。由于原始形式的损失函数以残差的对偶测试范数给出，核心目标之一是建立等效的可计算表达式。混合假设类在此发挥首要关键作用，其元素在（低维）时空变量中为分段多项式，而参数依赖的系数可通过神经网络等方式表示。其次，针对一阶SVF，我们区分两种情形：（i）测试空间可选为$L_2$空间（例如椭圆型或抛物型问题），此时残差位于$L_2$空间且可直接计算；（ii）当纤维问题的试探空间与测试空间（例如输运方程）依赖于参数时，我们采用超弱形式。结合间断Petrov-Galerkin方法，混合格式可有效推导出变分正确的可计算残差损失函数。我们的研究结果通过代表情形（i）和（ii）的数值实验加以说明，即具有分段常数扩散系数的椭圆边值问题，以及具有参数依赖对流场的纯输运方程。