Invariant-Net: Enhancing generalization in physics-informed neural networks via lie symmetry reduction

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving partial differential equations (PDEs). However, they fundamentally function as high-dimensional interpolators, often failing to capture the underlying causal dynamics required for out-of-distribution (OOD) extrapolation. In this work, we propose Invariant-Net, a geometrically grounded architecture that embeds Lie point symmetries directly into the learning process. Instead of approximating the solution in the original spatiotemporal domain, Invariant-Net projects the inputs onto a lower-dimensional quotient manifold determined by the symmetry group of the PDE. This effectively reduces the learning task from solving a high-dimensional PDE to approximating a simpler, often lower-dimensional, invariant function (e.g., a self-similar profile). We validate our framework on a diverse set of systems including the Porous Medium Equation (scaling symmetry), the Korteweg–de Vries equation (Galilean invariance), the 2D Heat equation (rotational symmetry), and the Nonlinear Schrödinger equation (internal phase symmetry). Our experiments demonstrate two critical advantages: (1) Dimensionality Reduction, which breaks the curse of dimensionality and significantly improves sample efficiency; and (2) Robust Generalization, where standard PINNs fail catastrophically. Specifically, in the time-extrapolation regime of the Porous Medium Equation, standard PINNs exhibit an error explosion from 28% to 75%, whereas Invariant-Net maintains a stable error of ≈5% indefinitely into the future. These results confirm that enforcing exact geometric symmetries allows neural networks to learn the intrinsic physical laws rather than merely fitting observed data.