Learning with Physical Constraints

TL;DR

This chapter presents three toy-problem tutorials on physics-constrained regression to emulate key fluid-dynamics challenges: (i) super-resolution and data assimilation for velocity fields via a constrained RBF regression enforcing boundary, divergence-free, and curl-free priors; (ii) data-driven turbulence closure learning Reynolds stresses from flow features within a 1D RANS framework, highlighting ill-posed regions and the need for physics-informed architecture; and (iii) adjoint-based parameter identification for time-dependent systems within a digital-twin context, demonstrated with a nonlinear pendulum. The approaches explore the benefits of integrating physical priors with machine learning while candidly addressing limitations such as ill-conditioned linear systems, vanishing gradients, and sensitivity to initial conditions. The chapter uses SPICY and a mixture of analytical and data-driven closures to illustrate both potential gains and practical hurdles in physics-constrained regression for fluid dynamics. Overall, it outlines a path toward robust hybrid modeling by combining accurate physics with targeted learning, while outlining concrete avenues for improvement and future research.

Abstract

This chapter provides three tutorial exercises on physics-constrained regression. These are implemented as toy problems that seek to mimic grand challenges in (1) the super-resolution and data assimilation of the velocity field in image velocimetry, (2) data-driven turbulence modeling, and (3) system identification and digital twinning for forecasting and control. The Python codes for all exercises are provided in the course repository.

Figures (13)

• Figure 1: Quiver plot of the training data in the constrained RBF regression exercise of Section \ref{['ch_3_sec_4']}.
• Figure 2: Training data and RBFs collocation from K-means clustering (left) and constraints and RBFs in constraint points (right).
• Figure 3: Resulting quiver plot after physics-constrained regression. The velocity field in the training data (left) and velocity field are on a regular grid due to super-resolution (right).
• Figure 4: Absolute error between RBF prediction and ground truth (left) and absolute error in the analytical curl field (right).
• Figure 5: Resulting errors in the prediction points. Absolute error between RBF prediction and ground truth (left) and absolute error in the analytical curl field. The error landscape is similar to the training data, and the model does not overfit.