Solving continuum and rarefied flows using differentiable programming

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.

Published in Journal of Computational Physics, Vol. 539 (2025), 114224. DOI: 10.1016/j.jcp.2025.114224

References in corpus (17)

• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• Solving high-dimensional partial differential equations using deep learning
• Efficient Deep Learning: A Survey on Making Deep Learning Models Smaller, Faster, and Better
• The Ensemble Kalman Filter for Inverse Problems
• Fast algorithms for computing the Boltzmann collision operator
• Ensemble Kalman Inversion: A Derivative-Free Technique For Machine Learning Tasks
• JAX-FLUIDS: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows
• Data-driven discovery of intrinsic dynamics
• Uniformly Accurate Machine Learning Based Hydrodynamic Models for Kinetic Equations
• Data-driven Discovery of Closure Models
• Learned discretizations for passive scalar advection in a 2-D turbulent flow
• Using neural networks to accelerate the solution of the Boltzmann equation
• Machine learning moment closure models for the radiative transfer equation I: directly learning a gradient based closure
• RelaxNet: A structure-preserving neural network to approximate the Boltzmann collision operator
• A stochastic kinetic scheme for multi-scale flow transport with uncertainty quantification
• Learning of viscosity functions in rarefied gas flows with physics-informed neural networks
• Hyperbolic Machine Learning Moment Closures for the BGK Equations