Estimating Parameter Fields in Multi-Physics PDEs from Scarce Measurements

arXiv:2509.00203v2 Announce Type: replace Abstract: Parameterized partial differential equations (PDEs) underpin the mathematical modeling of complex systems in diverse domains, including engineering, healthcare, and physics. A central challenge in using PDEs for real-world applications is to accurately infer the parameters, particularly when the parameters exhibit non-linear and spatiotemporal variations. Existing parameter estimation methods, such as sparse identification, physics-informed neural networks (PINNs), and neural operators, struggle in such cases, especially with nonlinear dynamics, multiphysics interactions, or limited observations of the system response. To address these challenges, we introduce Neptune, a general-purpose method capable of inferring parameter fields from sparse measurements of system responses. Neptune employs independent coordinate neural networks to continuously represent each parameter field in physical space or in state variables. Across various physical and biomedical problems, where direct parameter measurements are prohibitively expensive or unattainable, Neptune significantly outperforms existing methods, achieving robust parameter estimation from as few as 45 measurements, reducing parameter estimation errors by two orders of magnitude and dynamic response prediction errors by a factor of ten to baselines such as PINNs and neural operators. More importantly, it exhibits superior physical extrapolation capabilities, enabling reliable predictions in regimes far beyond the training data. By facilitating reliable and data-efficient parameter inference, Neptune promises broad transformative impacts in engineering, healthcare, and beyond.