Estimating Parameter Fields in Multi-Physics PDEs from Scarce Measurements

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.

Figures (6)

• Figure 1: Overview of Neptune, a generalizable method for inferring unknown parameter fields from sparse observations of system responses. Physical phenomena are often governed by known PDEs or coupled PDE systems with unknown key parameter fields. Neptune estimates these complex fields using extremely sparse physical response data through a two-stage process to ensure robustness. First, scalar parameters are estimated under an assumed homogeneous field approximation. Second, neural networks (referred to as $\mathcal{N}$) model the local variance of parameter fields, along with the previous scalar estimations. Both stages iteratively solve the PDE system numerically, minimizing the error between predicted and observed dynamics via adjoint-based backpropagation. The trained method yields a reliable physical system model capable of predicting phenomena under varying environmental conditions (e.g., source terms, boundary conditions), enabling Extreme Event Resilience, Accurate Extrapolation, and Proactive Strategies as demonstrated in predicting battery thermal runaway behavior, characterizing flow in porous media, and detecting cardiac fibrosis via electrophysiological analysis.
• Figure 2: Parameter estimation in cardiac electrophysiology.a. The training region with Gaussian noise during measurement, and the estimated forward inference of $V$. The predictions are compared between Neptune and PINN. b. The parameter estimation error under varying training data sizes. c. Comparison of parameter estimation error plots among Neptune, FNO, and PINN for two representative cases of $D$. d. The estimated forward response $V$ comparisons for case 1. When performing temporal extrapolation, PINN predicts values that do not adhere to physics laws.
• Figure 3: Parameter field estimation in thermal runaway problems.a. A prismatic battery, a cylinder battery, and random temperature measurements at the battery surface. Cross-sections $A-A$ and $B-B$ represent the cross-plane and the in-plane directions, respectively. b. The estimated three parameter fields $C_p, k(y-z)$, and $k(x)$ for the prismatic battery. c. Predicted thermal runaway behaviors with the estimated parameters. The initial temperature of the battery is 25°$C$ (or 298.15$K$), and the boundary condition (ambient temperature) is 423.15$K$. After accurately estimating the parameters, the model can correctly predict the thermal and chemical responses. d. Evaluation of temperature distribution for the prismatic battery at two typical times as marked in graph c. e. Predicted thermal runaway behaviors for the cylinder battery. f. Evaluation of temperature distribution for the cylinder battery.
• Figure 4: Parameter field estimation for flow in porous media.a. A demonstration of flow phenomena, and random measurements are taken as the ground truth for model training. b. The estimated hydraulic conductivity $K$ compared to the reference $K$. c. The estimated parameter field $K$ and its estimation error compared to the reference, for different numbers of training samples. The magnitudes are normalized. d. The particle concentration predictions (state responses) at different times. The predictions align well with the ground truth, even for extrapolation at $t=20$ minutes. e. The parameter estimation performance compared to different numbers of training samples. Error bands are derived from standard deviations from multiple estimations and represent the variability in the performance. f. The response prediction performance compared to different numbers of training samples.
• Figure 5: Parameter estimation in cell migration and proliferation.a. Parameter estimation results and comparisons. Neptune assumes all three parameters are density-dependent. The results show an increasing $\gamma$ value as density increases, while the other two parameter estimations exhibit minimal variation and can be considered as constants. This aligns well with assumptions from previous research sengers2007experimentalvo2015quantifying. b. Density response predictions. Using the estimated parameter fields, the density response at four different timesteps is displayed. Compared to results using the reference method (PINN-SR) and the scalar parameters setting, Neptune more accurately matches the experimental data at 36-hour and 48-hour, particularly in the regions with higher density values on both sides.