Equation identification for fluid flows via physics-informed neural networks

TL;DR

It is shown that a novel strategy that alternates between first- and second-order optimization proves superior to typical first-order strategies for estimating parameters and proposed a novel data-driven method to characterize PINN effectiveness in the inverse setting.

Abstract

Scientific machine learning (SciML) methods such as physics-informed neural networks (PINNs) are used to estimate parameters of interest from governing equations and small quantities of data. However, there has been little work in assessing how well PINNs perform for inverse problems across wide ranges of governing equations across the mathematical sciences. We present a new and challenging benchmark problem for inverse PINNs based on a parametric sweep of the 2D Burgers' equation with rotational flow. We show that a novel strategy that alternates between first- and second-order optimization proves superior to typical first-order strategies for estimating parameters. In addition, we propose a novel data-driven method to characterize PINN effectiveness in the inverse setting. PINNs' physics-informed regularization enables them to leverage small quantities of data more efficiently than the data-driven baseline. However, both PINNs and the baseline can fail to recover parameters for highly inviscid flows, motivating the need for further development of PINN methods.

Figures (8)

• Figure 1: We show a few temporal snapshots of the $u$-component of a Burgers' equation solution. At $t=0$, the solution exhibits variation only in the $y$-direction; as time progresses (e.g., for $t=0.14$ and $t=0.35$), rotational flow develops.
• Figure 2: For given quantities of data, we compare the estimation accuracy of (using Newton's method) and the data-driven strategy (\ref{['alg:estimation']}). Needing to minimize both the residual and data losses makes less effective at fitting the solution directly, yielding typically higher relative errors. However, the are generally better at fitting the parameters $\alpha$ and $\nu$.
• Figure 3: For varying amounts of training data, we plot the relative errors in estimating the Burgers' solution (top), convection coefficient $\alpha$ (middle) and diffusion coefficient $\nu$ (bottom), using the data-driven (\ref{['alg:estimation']}) strategy across different Burgers' parameters. The black dashed line indicates $10\%$ or less error, our threshold for success. With $131072$ points, the can achieve less than $10\%$ solution relative error for every parameter configuration. However, even with $131072$ data points, it struggles to estimate parameters, failing at estimating $\alpha$ for every configuration other than $\nu=0.01$ and failing at estimating $\nu$ in five out of the ten parameter configurations.
• Figure 4: For varying amounts of training data (columns) and different optimizers (rows), we plot the relative errors in estimating the diffusion coefficient $\nu$ (left) and convection coefficient $\alpha$ (right), using . The black dashed line indicates $10\%$ or less error, our threshold for success. Compared to the data-driven baseline (\ref{['fig:baseline_results']}), are more successful in recovering parameters at a given quantity of labeled training data. They can correctly estimate $\nu$ across values of $\alpha$ except when $\nu=0.0001$ (i.e., when the fluid is highly inviscid). They are also successful in estimating $\alpha$, except in the $\nu=0.01$ regime.
• Figure 5: For varying amounts of training data (columns) and different optimizers (rows), we plot the relative errors in the predicted solutions to the Burgers' equation.