182 citations · 616 across the 20 of their papers we have counts for
13 papers · 1 filter
Trust Region Method for Coupled Systems of PDE Solvers and Deep Neural Networks
Kailai Xu, Eric Darve
Physics-informed machine learning and inverse modeling require the solution of ill-conditioned non-convex optimization problems. First-order methods, such as SGD and ADAM, and quas…
ADCME: Learning Spatially-varying Physical Fields using Deep Neural Networks
Kailai Xu, Eric Darve
ADCME is a novel computational framework to solve inverse problems involving physical simulations and deep neural networks (DNNs). This paper benchmarks its capability to learn spa…
Solving Inverse Problems in Steady-State Navier-Stokes Equations using Deep Neural Networks
Tiffany Fan, Kailai Xu, Jay Pathak +1
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 genera…
Inverse Modeling of Viscoelasticity Materials using Physics Constrained Learning
Kailai Xu, Alexandre M. Tartakovsky, Jeff Burghardt +1
We propose a novel approach to model viscoelasticity materials using neural networks, which capture rate-dependent and nonlinear constitutive relations. However, inputs and outputs…
Learning Constitutive Relations using Symmetric Positive Definite Neural Networks
Kailai Xu, Daniel Z. Huang, Eric Darve
We present the Cholesky-factored symmetric positive definite neural network (SPD-NN) for modeling constitutive relations in dynamical equations. Instead of directly predicting the…
Physics Constrained Learning for Data-driven Inverse Modeling from Sparse Observations
Kailai Xu, Eric Darve
Deep neural networks (DNN) have been used to model nonlinear relations between physical quantities. Those DNNs are embedded in physical systems described by partial differential eq…
Learning Hidden Dynamics using Intelligent Automatic Differentiation
Kailai Xu, Dongzhuo Li, Eric Darve +1
Many engineering problems involve learning hidden dynamics from indirect observations, where the physical processes are described by systems of partial differential equations (PDE)…
Adversarial Numerical Analysis for Inverse Problems
Kailai Xu, Eric Darve
Many scientific and engineering applications are formulated as inverse problems associated with stochastic models. In such cases the unknown quantities are distributions. The appli…
Learning Constitutive Relations from Indirect Observations Using Deep Neural Networks
Daniel Z. Huang, Kailai Xu, Charbel Farhat +1
We present a new approach for predictive modeling and its uncertainty quantification for mechanical systems, where coarse-grained models such as constitutive relations are derived…
The Neural Network Approach to Inverse Problems in Differential Equations
Kailai Xu, Eric Darve
We proposed a framework for solving inverse problems in differential equations based on neural networks and automatic differentiation. Neural networks are used to approximate hidde…
Spectral Method for the Fractional Laplacian in 2D and 3D
Kailai Xu, Eric Darve
A spectral method is considered for approximating the fractional Laplacian and solving the fractional Poisson problem in 2D and 3D unit balls. The method is based on the explicit f…
Efficient Numerical Method for Models Driven by Lévy Process via Hierarchical Matrices
Kailai Xu, Eric Darve
Modeling via fractional partial differential equations or a Lévy process has been an active area of research and has many applications. However, the lack of efficient numerical com…
Isogeometric Collocation Method for the Fractional Laplacian in the 2D Bounded Domain
Kailai Xu, Eric Darve
We consider the isogeometric analysis for fractional PDEs involving the fractional Laplacian in two dimensions. An isogeometric collocation method is developed to discretize the fr…