Solving continuum and rarefied flows using differentiable programming

TL;DR

The paper tackles the challenge of simulating multi-scale flows that span continuum and rarefied regimes by formulating a fully differentiable simulator based on differentiable programming. It leverages adjoint sensitivity and automatic differentiation to enable end-to-end optimization of parametric mechanical and neural components within forward flow solvers, unifying CFD and machine learning workflows. Across a suite of experiments, the approach demonstrates flux-parameter optimization, viscosity identification, neural-closure construction, and DeepONet-based operator learning for the Boltzmann equation, achieving high accuracy with substantial computational efficiency gains. The work provides open-source code under MIT license and outlines a path toward physics-informed data-driven discovery, surrogate modeling, and acceleration for cross-scale gas dynamics, with potential extensions to radiative transfer, plasma, and stochastic simulations.

Abstract

Accurate and efficient prediction of multi-scale flows remains a formidable challenge. Constructing theoretical models and numerical methods often involves the design and optimization of parameters. While gradient descent methods have been mainly manifested to shine in the wave of deep learning, composable automatic differentiation can advance scientific computing where the application of classical adjoint methods alone is infeasible or cumbersome. Differentiable programming provides a novel paradigm that unifies data structures and control flows and facilitates gradient-based optimization of parameters in a computer program. This paper addresses the notion and implementation of the first solution algorithm for multi-scale flow physics across continuum and rarefied regimes based on differentiable programming. The fully differentiable simulator provides a unified framework for the convergence of computational fluid dynamics and machine learning, i.e., scientific machine learning. Specifically, parameterized mechanical-neural flow models and numerical methods can be constructed for forward physical processes, while the parameters can be trained on the fly with the help of the gradients that are taken through the backward passes of the whole simulation program, a.k.a., end-to-end optimization. As a result, versatile data-driven modeling and simulation can be achieved for physics discovery, surrogate modeling, and simulation acceleration. The fundamentals and implementation of the solution algorithm are demonstrated in detail. Numerical experiments, including forward and inverse problems for hydrodynamic and kinetic equations, are presented to demonstrate the performance of the numerical method. The open-source codes to reproduce the numerical results are available under the MIT license.

Figures (15)

• Figure 1: Schematic of forward-mode automatic differentiation for a sequence of functions.
• Figure 2: Schematic of reverse-mode automatic differentiation for a sequence of functions.
• Figure 3: Flow of the solution and training algorithms for the unified mechanical-neural model based on differentiable programming.
• Figure 4: Architecture of DeepONet surrogate model of the Boltzmann equation.
• Figure 5: Profiles of density, $U$-velocity, temperature, and pressure at $t=0.2$ simulated by the single-parameter model in the Sod shock tube problem.