Estimating Parameter Fields in Multi-Physics PDEs from Scarce Measurements

2025-08-29

AI Summary

To address the challenge of accurately inferring nonlinear, spatiotemporally varying parameter fields in multiphysics partial differential equations from sparse measurement data, this paper proposes Neptune—a novel physics-informed inversion framework. Neptune employs Independent Coordinate Neural Networks (ICNNs) to represent parameter fields as continuous, decoupled implicit functions, thereby circumventing the expressivity limitations and poor generalizability inherent in conventional grid-based or parametric representations. Integrated within the Physics-Informed Neural Networks (PINNs) paradigm, Neptune jointly optimizes the parameter field and system dynamics under rigorous physical constraints. Evaluated on benchmark multiphysics problems, Neptune achieves over two orders-of-magnitude reduction in parameter estimation error and a tenfold decrease in dynamic response prediction error—using only 50 sparse observations. Moreover, it demonstrates exceptional extrapolation accuracy in unseen spatiotemporal domains, substantially outperforming state-of-the-art inversion methods.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 and physics-informed neural networks (PINNs), 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 50 observations, reducing parameter estimation errors by two orders of magnitude and dynamic response prediction errors by a factor of ten compared to PINNs. Furthermore, Neptune exhibits superior extrapolation capabilities, enabling accurate predictions in regimes beyond training data where PINN fail. By facilitating reliable and data-efficient parameter inference, Neptune promises broad transformative impacts in engineering, healthcare, and beyond.

Problem

• Estimating nonlinear spatiotemporal parameter fields in multi-physics PDEs
• Overcoming limitations of scarce system response measurements
• Addressing multiphysics interactions where traditional methods fail

Innovation

• Independent coordinate networks represent parameter fields
• Robust estimation from scarce observations via neural networks
• Superior extrapolation beyond training data compared PINNs