Sujal Dave, PhD

Sujal Dave, PhD

Physics-Constrained DeepONet for Surrogate CFD Models

Overview

This project introduces a Physics-Constrained DeepONet (PC-DeepONet) architecture designed to create highly efficient surrogate models for computational fluid dynamics (CFD). As detailed in the article “76_Physics_constrained_DeepONe.pdf” presented at the ICLR 2024 Workshop on AI4Differential Equations In Science, the framework embeds the fundamental physics knowledge of the continuity equation directly into the neural network’s structure. The model learns the operator mapping from the parameterized geometry of a curved backward-facing step (BFS) to the resulting velocity and pressure fields.

The curved BFS geometry is parameterized using a third-degree (cubic) non-uniform rational B-spline (NURBS). By fixing the boundary control points and varying the x-coordinates of two intermediate control points between (-0.25) and (1.25), a dataset of 50 distinct slope shapes was generated. High-fidelity CFD training data was simulated using FreeFEM++ at a Reynolds number of (1000) on 2D unstructured meshes containing approximately 50,000 elements each. Unlike a purely data-driven vanilla DeepONet, the PC-DeepONet strictly enforces a divergence-free velocity field by constructing a skew-symmetric matrix from the Jacobian of the network’s output and applying the divergence operator.

Results

The PC-DeepONet successfully learned the nonlinear operator mapping from the NURBS control point parameters to the flow field variables (U, V, and P) using sparse training data consisting of only 1000 randomly sampled spatial sensors.

  • The physics-constrained architecture achieved a relative L2 error of (4.45 \times 10^{-3}) and a validation loss of (1.82 \times 10^{-6}). This significantly outperformed the baseline vanilla DeepONet, which yielded a relative L2 error of (1.16 \times 10^{-2}) and a validation loss of (1.23 \times 10^{-5}).

  • Demonstrating high computational efficiency, the PC-DeepONet converged in just 50 iterations using the L-BFGS optimizer on the small 50-geometry dataset.

  • Leveraging the resolution-free capability of neural operators, the trained PC-DeepONet effectively extrapolated the flow fields onto unseen dense meshes of 50,000 points.

  • Limitations & Future Work: Absolute error analysis revealed that discrepancies primarily clustered near the boundary layers. This is attributed to under-sampling near the variable slope surface where points cannot be uniformly mapped across a common mesh. Future work will focus on integrating hard boundary constraints and masking approaches to improve near-wall performance for more complex, turbulent flows.