Solving Inverse Problems in Steady-State Navier-Stokes Equations using Deep Neural Networks

Abstract

Inverse problems in fluid dynamics are ubiquitous in science and engineering, with applications ranging from electronic cooling system design to ocean modeling. We propose a general and robust approach for solving inverse problems for the steady state Navier-Stokes equations by combining deep neural networks and numerical PDE schemes. Our approach expresses numerical simulation as a computational graph with differentiable operators. We then solve inverse problems by constrained optimization, using gradients calculated from the computational graph with reverse-mode automatic differentiation. This technique enables us to model unknown physical properties using deep neural networks and embed them into the PDE model. Specifically, we express the Newton's iteration to solve the highly nonlinear Navier-Stokes equations as a computational graph. We demonstrate the effectiveness of our method by computing spatially-varying viscosity and conductivity fields with deep neural networks (DNNs) and training DNNs using partial observations of velocity fields. We show that DNNs are capable of modeling complex spatially-varying physical field with sparse and noisy data. We implement our method using ADCME, a library for solving inverse modeling problems in scientific computing using automatic differentiation.

Hybrid FEM-NN models: Combining artificial neural networks with the finite element method

“…Although the error in the solution u is actually lower than for the hybrid method, the pointwise estimator produces a non-smooth solution to the inverse problem. This is in line with previous results in [9,12], where it is shown that the neural network can act as regularisation and produces smooth approximations.…”

“…the geometric flexibility and rich set of finite element functions of the finite element method, with the flexibility of neural networks to express unknown functions. Equation discovery using traditional PDE solvers in conjunction with neural networks has previously been demonstrated [9,10,11,12]. In [9] a neural network with a single hidden layer is used to approximate a spatially varying diffusion coefficient in the Poisson equation that was discretised with the finite element method.…”

“…While the framework has been demonstrated on learning constitutive relations and physical parameters in the steady state Navier-Stokes equation [11,12], it (currently) lacks support for time-dependent problems.…”

Monarch: Expressive Structured Matrices for Efficient and Accurate Training

“…Deep learning has found application in the domain of differential equations and scientific computing [85], with methods developed for prediction and control problems [56,71], as well as acceleration of numerical schemes [83,51]. Specific to the partial differential equations (PDEs) are approaches designed to learn solution operators [87,29,65], and hybridized solvers [59], evaluated primarily on classical fluid dynamics.…”

Physics-consistent deep learning for structural topology optimization

“…Although deep learning has many diverse applications and has demonstrated extraordinary results in several real-world scenarios, our focus in this paper is the recent application of deep learning to learn a system's underlying physics. There has been an increased interest in learning physical phenomena with neural networks in order to reduce the data requirement and achieve better performance with very little or no data 4,9,10,12,15,[18][19][20][21]23,27,31,32 . One method by which this can be achieved is by modifying the loss function Using the DLTO framework, we predict the optimal density of the geometry without any requirement of iterative finite element evaluations.…”