PIED: Physics-Informed Experimental Design for Inverse Problems

TL;DR

PieD tackles learning informative, one-shot experimental designs for inverse PDE parameter problems by marrying physics-informed neural networks (PINNs) with differentiable, gradient-based design optimization. It introduces a fully differentiable ED loop where PINNs serve as both forward simulators and inverse solvers, enabling continuous optimization of the observation input $X$ under budget constraints. A shared meta-learned initialization accelerates training across multiple PDE parameters, and two scalable criteria, Few-step Inverse Solver Training (FIST) and Model Training Estimate (MoTE), approximate inverse solver performance to guide design without full IP convergence. Empirical results on finite- and function-valued-parameter PDEs, plus real-data cases (groundwater and cell growth), show PIED outperforms baselines and leverages PINN advantages to reduce data collection costs while maintaining or improving IP accuracy.

Abstract

In many science and engineering settings, system dynamics are characterized by governing PDEs, and a major challenge is to solve inverse problems (IPs) where unknown PDE parameters are inferred based on observational data gathered under limited budget. Due to the high costs of setting up and running experiments, experimental design (ED) is often done with the help of PDE simulations to optimize for the most informative design parameters to solve such IPs, prior to actual data collection. This process of optimizing design parameters is especially critical when the budget and other practical constraints make it infeasible to adjust the design parameters between trials during the experiments. However, existing experimental design (ED) methods tend to require sequential and frequent design parameter adjustments between trials. Furthermore, they also have significant computational bottlenecks due to the need for complex numerical simulations for PDEs, and do not exploit the advantages provided by physics informed neural networks (PINNs), such as its meshless solutions, differentiability, and amortized training. This work presents PIED, the first ED framework that makes use of PINNs in a fully differentiable architecture to perform continuous optimization of design parameters for IPs for one-shot deployments. PIED overcomes existing methods' computational bottlenecks through parallelized computation and meta-learning of PINN parameter initialization, and proposes novel methods to effectively take into account PINN training dynamics in optimizing the ED parameters. Through experiments based on noisy simulated data and even real world experimental data, we empirically show that given limited observation budget, PIED significantly outperforms existing ED methods in solving IPs, including challenging settings where the inverse parameters are unknown functions rather than just finite-dimensional.

Figures (11)

• Figure 1: Comparison between observation selection and solving IPs in real life (\ref{['fig:pied-inv']}), versus the proess as modelled in the PIED framework (\ref{['fig:pied-loop']}).
• Figure 2: Results for meta-learning a shared NN initialization for PINNs trained on 1D damped oscillator case. The shared initialization (blue line) in \ref{['fig:si-shared']} exhibits similar structure to PDE solutions $u_\beta$ for different values of $\beta$ (faint green lines), unlike the random initialization (blue line) in \ref{['fig:si-rand']}. In \ref{['fig:si-for', 'fig:si-inv']}, we show that this translates to better average train and test loss performance of PINNs with shared initialization compared to random initialization w.r.t. different $\beta$.
• Figure 3: Results for learning a NN initialization for PINNs trained on Eikonal equation case. \ref{['fig:sieik-rand']} represents a randomly initialized PINN, while \ref{['fig:sieik-shared']} represents the shared initialzation for the PINN. \ref{['fig:sieik-samplesoln']} shows sample PDE solutions $u_\beta$ for different random PDE parameters $\beta$.
• Figure 4: Example of ED process involving the Eikonal equation using different methods of observation selection. Top row: the true observation $T(x, y)$ and its approximation via PINNs. Middle row: the true unknown function $v(x, y)$ and the recovered estimations based on observation inputs. Bottom row: the error of the reconstructed $v(x, y)$.
• Figure 5: Visualization of real-life experimental data used in our tests, along with demonstration of observations selected by FIST. \ref{['fig:gw']}: example of groundwater flow data from shadab_investigating_2023. Gray points represent the collected data, while the blue points are the observations chosen by FIST. The black line represents the prediction from the corresponding inverse PINN. \ref{['fig:cellpop']}: example of cell population growth data from jin_reproducibility_2016. The left figure shows the cell population data which are collected at 12 hour intervals, while the right figure shows the population prediction from the inverse PINN. In both figures, the blue points represent the observations chosen by FIST.