CertINR: Certified Verification and Compression of Implicit Neural Fields

Implicit neural representations (INRs) use neural networks to represent continuous functions over spatial or spatiotemporal coordinates. After compression, the decoded network can differ from the reference in ways that evaluations at sampled coordinates may miss. We introduce CertINR, a framework with two coupled components: a verifier that provides provable upper bounds on the maximum reference–decoded error over the continuous domain, and a certificate-aware compressor that uses these bounds to repair and select compressed representations. The key verification idea is to analyze the reference and decoded networks jointly at the same coordinates, so that variation shared by the two networks cancels before their difference is bounded. We realize this idea through paired Taylor bounds that preserve this cancellation through nonlinear layers and admit refinement-based certification guarantees. Alongside paired Taylor, we develop complementary global spectral and local residual/Jacobian certificates and combine all three in a verification portfolio. On the compression side, CertINR searches a pool of compressed candidates, uses the local error models to repair the final affine layer, verifies the actual decoded streams, and returns the smallest representation that satisfies the target error tolerance. Experiments on analytic and pretrained sinusoidal fields show that no single certificate dominates across network regimes: paired Taylor is strongest on shallow analytic models, whereas spectral bounds are stronger on larger pretrained models. The CertINR bounds also resolve cases left uncertified by general-purpose neural-network verifiers. On six pretrained signed distance fields (SDFs), CertINR achieves 41.21% mean per-shape storage savings at tolerance 0.01. More generally, we show that for SDFs, a certified uniform error preserves inside–outside decisions and, under explicit gradient conditions, can also guarantee topology preservation and bound surface displacement.