UC Santa Barbara

Two directions of utilizing machine learning techniques in computational mathematics are explored with case studies on two real-world projects.

The first direction is in the supervised learning category where data generated from traditional numerical solvers are leveraged to train a machine learner for fast prediction or other specific task, like an inverse problem. A new framework that leverages data obtained from self-consistent field theory (SCFT) simulations with deep learning to accelerate the exploration of parameter space for block copolymers is presented in the first project. Deep neural networks are adapted and trained in Sobolev space to better capture the saddle point nature of the SCFT approximation. The proposed approach consists of two main problems: 1) the learning of an approximation to the effective Hamiltonian as a function of the average monomer density fields and the relevant physical parameters and 2) the prediction of saddle density fields given the polymer parameters. There is an additional challenge: the effective Hamiltonian has to be invariant under shifts (and rotations in 2D and 3D). Two different methods are developed to achieve the invariance: 1) a data-enhancing approach and an appropriate regularization and 2) a designed convolutional neural network (CNN). Generative adversarial networks (GAN) are connected with the SCFT models to predict the initial saddle density fields. These fields are selected from either GAN generations or the training set, and are fine-tuned by the deep Sobolev space-trained neural networks to get the final prediction.

The second direction is in the unsupervised learning category where machine learning techniques are developed to solve a previously unsolvable system. The proposed new approach embeds deep neural networks inside numerical schemes, builds suitable objective functions, and determines the neural network parameters by minimizing the objective functions. As a case study, a multi-region SEIR (Susceptible - Exposed - Infectious - Recovered) model based on stochastic differential game theory is proposed. This model's aim is to aid in the formulation of optimal regional policies for infectious diseases. Specifically, the standard epidemic SEIR model is enhanced by taking into account the social and health policies issued by multiple region planners. This enhancement makes the model more realistic and powerful. However, it also introduces a formidable computational challenge due to the high dimensionality of the solution space brought by the presence of multiple regions. This significant numerical difficulty of the model structure motivates a generalization of the deep fictitious algorithm introduced in [Han and Hu, MSML2020, pp.221--245, PMLR, 2020]. As a result, an improved algorithm, which uses deep neural networks to approximate the value function and control parameters and that builds objective functions based on the error of the approximated value function at the terminal time is developed. The proposed extended model and the new deep learning-based algorithm are applied to study the COVID-19 pandemic in three states: New York, New Jersey and Pennsylvania. The results show the effects of the lockdown/travel ban policy on the spread of COVID-19 for each state and how their policies affect each other.