NeuralDMD: Interpretable Untrained Neural Network for Imaging from Sparse and Noisy Observations

Many challenges in scientific imaging involve solving ill-posedinverse problems, where the goal is to recover spatio-temporal fields fromindirect, noisy, and highly sparse measurements — often without accessto ground truth data or reliable simulators. To address this challengingscenario, we present NeuralDMD, an interpretable, untrained (per-instance) reconstruction framework that combines neural implicit rep-resentations with Dynamic Mode Decomposition (DMD) to reconstructcontinuous spatio-temporal dynamics directly from measurements. Neu-ralDMD parameterizes DMD modes as continuous neural fields, andimposes a low-rank linear dynamics prior with spectral time evolutionto enforce temporal continuity. This formulation enables both forecast-ing under sparsity, and yields interpretable modes and spectra. We findthat NeuralDMD outperforms baselines on a wide variety of tasks: fromweather data assimilation from sparse station observations to interfero-metric (Fourier domain) observations of Sagittarius A*, the black hole atthe center of our galaxy. Moreover, NeuralDMD remains stable whenextrapolating into the future. While this framework is most naturallysuited to linear dynamics, we show that it can be applied to nonlinearregimes, though with extrapolation performance that degrades with in-creasing nonlinearity. Together, these results show that NeuralDMDenables interpretable reconstruction and forecasting of spatio-temporaldynamics directly from sparse and indirect measurements without rely-ing on numerical simulators or training data.