Tensor-Network Finite Elements for Analytic Operator Equations

Operator equations (OEs) underpin quantitative modeling across science and engineering. Finite-element (FE) methods discretize continuous OEs into finite-dimensional algebraic systems, whereas tensor networks (TNs) provide flexible variational representations of correlated discrete systems. Here, we develop a framework that connects FE with TN for analytic OEs. The power of this method comes from its ability to convert highly non-linear partial differential equations into linear matrix equations. In particular, we show that FE discretization induces a hierarchy of multilinear interaction tensors, through which differential, integral, nonlinear, memory, and delay equations can be expressed within a common algebraic structure. The resulting systems are reformulated as weighted-residual optimization problems over TN degrees of freedom. Matrix-product-state calculations for one-dimensional linear and nonlinear diffusion reproduce conventional solutions with controlled error while preserving continuity and Neumann boundary conditions. The framework provides a common variational language for analytic OEs and establishes a direct connection between FE numerical formalism and TN variational algorithms, offering a general foundation for TN-based and quantum-inspired approaches to solving OEs.