Certified Neural PDE Solvers: Constructive Verification of Physics-Informed Networks via Dependent Type Theory

Keywords: Physics-informed neural networks, PINNs, dependent type theory, formal verification, Lean 4, certified computation, PDEs, a-posteriori error estimation, constructive mathematics

TL;DR: CertPINN uses dependent types to produce machine-checkable certificates that a PINN solution satisfies the PDE within a given tolerance.

Abstract: Physics-informed neural networks (PINNs) have demonstrated remarkable empirical success in solving partial differential equations, yet their solutions fundamentally lack formal correctness guarantees. We present **CertPINN**, the first framework for constructively verifying PINN solutions using dependent type theory. Our approach formalizes PDE residual bounds within a constructive logic framework, producing machine-checkable certificates that a neural network output satisfies the governing equations within a specified tolerance. Key contributions include: (1) a type-theoretic formalization of weak PDE solutions where solution types encode both function spaces and error bounds; (2) a constructive proof that PINN training with our certified loss function converges to a solution within $\varepsilon$ of the true weak solution in $H^1$ norm, verified end-to-end in Lean 4; (3) a certified a-posteriori error estimator providing pointwise error bounds without knowledge of the true solution; (4) experimental validation on five benchmark PDEs (Poisson, heat, Burgers, Navier-Stokes 2D, Schrödinger) demonstrating that CertPINN produces verified solutions with only 3-7\% computational overhead compared to standard PINNs, while guaranteeing error bounds with mathematical certainty rather than empirical confidence.

Submission Number: 146