Tensor-Train Compressed Separable PINNs: A Curvature-Aware Optimization Framework for Parametric PDEs in High Dimensions

Denis Korolev, Martin Eigel

In this work, we develop a second-order optimization framework for physics-informed neural networks (PINNs) applied to high-dimensional parametric partial differential equations (PDEs). The framework is built on the Gauss--Newton pullback metric, which provides an operator-informed notion of curvature in parameter space and connects the method to the broader family of natural gradient schemes. We show that, for coordinate-separable neural architectures and linear differential operators (or linearized operators in the nonlinear case) admitting a finite separable representation, the residual Jacobian inherits a structured separable factorization. This yields an exact compressed formulation of the Gauss--Newton step in a reduced space, without assembling the full residual Jacobian on the exponentially large tensor-product collocation grid. The dimension of the reduced space (the effective compressed dimension) is determined by the local collocation grid sizes, the separable operator structure, and the contraction pattern of the architecture, thereby avoiding dependence on the full tensor-product grid size and replacing dense linear algebra in parameter space by a substantially…