Hard-constrained Physics-informed Neural Networks for Interface Problems

Abstract: Physics-informed neural networks (PINNs) have emerged as a flexible framework for solving partial differential equations, but their performance on interface problems remains challenging because continuity and flux conditions are typically imposed through soft penalty terms. The standard soft-constraint formulation leads to imperfect interface enforcement and degraded accuracy near interfaces. We introduce two ansatz-based hard-constrained PINN formulations for interface problems that embed the interface physics into the solution representation and thereby decouple interface enforcement from PDE residual minimization. The first, termed the windowing approach, constructs the trial space from compactly supported windowed subnetworks so that interface continuity and flux balance are satisfied by design. The second, called the buffer approach, augments unrestricted subnetworks with auxiliary buffer functions that enforce boundary and interface constraints at discrete points through a lightweight correction. We study these formulations on one- and two-dimensional elliptic interface benchmarks and compare them with soft-constrained baselines. In one-dimensional problems, hard constraints consistently improve interface fidelity and remove the need for loss-weight tuning; the windowing approach attains very high accuracy (as low as $O(10<sup>{-9})$) on simple structured cases, whereas the buffer approach remains accurate ($\sim O(10<sup>{-5})$) across a wider range of source terms and interface configurations. In two dimensions, the buffer formulation is shown to be more robust because it enforces constraints through a discrete buffer correction, as the windowing construction becomes more sensitive to overlap and corner effects and over-constrains the problem. This positions the buffer method as a straightforward and geometrically flexible approach to complex interface problems.