10:30AM - 12:10AM

Efficient and accurate structure preserving schemes for a class of complex nonlinear systems

Jie Shen

Purdue University

Abstract

We present in this talk the scalar auxiliary variable (SAV) approach to deal with nonlinear terms in a large class of complex dissipative/conservative systems. In particular, for gradient flows driven by a free energy, it leads to linear and unconditionally energy stable second-order (extendable to higher-orders) schemes which only require solving decoupled linear equations with constant coefficients. Hence, these schemes are extremely efficient as well as accurate, which are also validated by ample numerical results. We shall present a convergence and error analysis under mild assumptions on the nonlinear free energy, and discuss applications of the SAV approach to various complex dissipative/conservative systems.

Diffuse domain methods for solving PDEs in complex geometries

John S. Lowengrub

University of California, Irvine

Abstract

The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness ε that is typically proportional to the minimum grid size. The original PDE is reformulated in a larger, regular domain using a smoothed characteristic function of the complex domain and singular source terms are introduced to approximate the boundary conditions. The reformulated equation, which is independent of the dimension and domain geometry, can be solved by standard numerical methods and the same solver can be used for any domain geometry, which could be dynamically-evolving. A challenge is making the method higher-order accurate. Current implementations demonstrate a wide range in their accuracy but often yield at best first order accuracy in ε. In this talk, we analyze the diffuse-domain PDEs using matched asymptotic expansions and explain the observed behaviors. Our analysis also identifies simple modifications to the diffuse-domain PDEs that yield higher-order accuracy in ε. Our analytic results are confirmed numerically and we present examples from materials science, fluid dynamics and the life sciences that demonstrate the accuracy and utility of this methods.

Random Batch Methods for Interacting Particle Systems

Shi Jin

Shanghai Jiao Tong University

Abstract

We develop random batch methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. On one hand, these methods are efficient Asymptotic-Preserving schemes for the underlying particle systems, allowing N-independent time steps and also capture, in the N \to \infty limit, the solution of the mean field limit which are nonlinear Fokker-Planck equations; on the other hand, the stochastic processes generated by the algorithms can also be regarded as new models for the underlying problems. For one of the methods, we give a particle number independent error estimate under some special interactions. Then, we apply these methods to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems. This is a joint work with Lei Li (Shanghai Jiao Tong University) and Jian-Guo Liu (Duke University).

A thermodynamically consistent phase field crystal model with heat transport

Steven Wise

University of Tennessee

Abstract

In this talk, I will describe some preliminary work on a thermodynamically consistent phase field crystal model with multiple structural phases and heat transport. The model is based on local and global entropy production. The model is applied to the problems of crystalline solidification and graphene growth on a substrate via vapor deposition.

Uncertainty Quantification and Machine Learning for learning the Physical Laws and predicting heterogeneous elliptic PDE solutions on varied domains

Guang Lin

Purdue University

Abstract

In this talk, I will present a new data-driven paradigm on how to quantify the structural uncertainty (model-form uncertainty) and learn the physical laws hidden behind the noisy data in the complex systems governed by partial differential equations. The key idea is to identify the terms in the underlying equations and to approximate the coefficients of the terms with error bars using Bayesian machine learning algorithms on the available noisy measurement. In particular, Bayesian sparse feature selection and parameter estimation are performed. Numerical experiments show the robustness of the learning algorithms with respect to noisy data and size, and its ability to learn various candidate equations with error bars to represent the quantified uncertainty.

Next, I will introduce a framework for constructing light-weight numerical solvers for partial differential equations (PDEs) using convolutional neural networks. A theoretical justification for the neural network approximation to partial differential equation solvers on varied domains is established based on the existence and properties of Green's functions. These solvers are able to effectively reduce the computational demands of traditional numerical methods into a single forward-pass of a convolutional network. The network architecture is also designed to predict pointwise Gaussian posterior distributions, with weights trained to minimize the associated negative log-likelihood of the observed solutions. This setup facilitates simultaneous training and uncertainty quantification for the network's solutions, allowing the solver to provide pointwise uncertainties for its predictions. The associated training procedure avoids the computationally expensive Bayesian inference steps used by other state-of-the-art uncertainty models and allows training to be scaled to the large data sets required for learning on varied problem domains. The performance of the framework is demonstrated on three distinct classes of PDEs consisting of two linear elliptic problem setups and a nonlinear Poisson problem. After a single offline training procedure for each class, the proposed networks are capable of accurately predicting the solutions to linear and nonlinear elliptic problems with heterogeneous source terms defined on any specified two-dimensional domain using just a single forward-pass of a convolutional neural network. Additionally, an analysis of the predicted pointwise uncertainties is presented with experimental evidence establishing the validity of the network's uncertainty quantification schema.

Some deep learning strategies for mechanics and physics

Ankit Srivastava

Illinois Institute of Technology

Abstract

In this talk, we present the application of deep neural networks to two problems of mechanics/physics. The first problem is the prediction of the vibrational eigenvalues of phononic crystals. We show that Convolutional Neural Networks (CNNs) can be used to effectively learn the highly nonlinear relationship between problem parameters and eigenvalues. With the right dataset, CNNs generalize well to unseen cases and demonstrate prediction errors which are consistently below traditional neural network architectures even when the former is strained on a fraction of the data that the latter is trained on. The second problem is the case of approximating underlying physical laws by analyzing the associated physical phenomenon. Here we show that the principle of spatio-temporal locality, inherent in all physical laws, can be used to create highly data efficient deep learning architectures capable of embodying/discovering various elements of physical laws. We propose that these networks -- termed Physics Constrained Deep Networks (PCDN) -- can efficiently approximate boundary conditions, constitutive relations, and/or governing equations of physical system and then predict the evolution of the system under unseen conditions. As a specific example, networks are designed to understand and predict the time evolution of a system of colliding balls in a box in a gravity field. We show that learning from just one simulation, our networks are able to predict the evolution of unseen systems characterized by different number of balls, different sized/shaped box etc.