Xiangting Li

Mingtao Xia, Xiangting Li, Qijing Shen et al.

Rapidly developing machine learning methods has stimulated research interest in computationally reconstructing differential equations (DEs) from observational data which may provide additional insight into underlying causative mechanisms. In this paper, we propose a novel neural-ODE based method that uses spectral expansions in space to learn spatiotemporal DEs. The major advantage of our spectral neural DE learning approach is that it does not rely on spatial discretization, thus allowing the target spatiotemporal equations to contain long range, nonlocal spatial interactions that act on unbounded spatial domains. Our spectral approach is shown to be as accurate as some of the latest machine learning approaches for learning PDEs operating on bounded domains. By developing a spectral framework for learning both PDEs and integro-differential equations, we extend machine learning methods to apply to unbounded DEs and a larger class of problems.

Mingtao Xia, Xiangting Li, Qijing Shen et al.

We analyze the Wasserstein distance ($W$-distance) between two probability distributions associated with two multidimensional jump-diffusion processes. Specifically, we analyze a temporally decoupled squared $W_2$-distance, which provides both upper and lower bounds associated with the discrepancies in the drift, diffusion, and jump amplitude functions between the two jump-diffusion processes. Then, we propose a temporally decoupled squared $W_2$-distance method for efficiently reconstructing unknown jump-diffusion processes from data using parameterized neural networks. We further show its performance can be enhanced by utilizing prior information on the drift function of the jump-diffusion process. The effectiveness of our proposed reconstruction method is demonstrated across several examples and applications.

Joshua C Chang, Xiangting Li, Shixin Xu et al.

Importance sampling (IS) allows one to approximate leave one out (LOO) cross-validation for a Bayesian model, without refitting, by inverting the Bayesian update equation to subtract a given data point from a model posterior. For each data point, one computes expectations under the corresponding LOO posterior by weighted averaging over the full data posterior. This task sometimes requires weight stabilization in the form of adapting the posterior distribution via transformation. So long as one is successful in finding a suitable transformation, one avoids refitting. To this end, we motivate the use of bijective perturbative transformations of the form $T(\boldsymbolθ)=\boldsymbolθ + h Q(\boldsymbolθ),$ for $0<h\ll 1,$ and introduce two classes of such transformations: 1) partial moment matching and 2) gradient flow evolution. The former extends prior literature on moment-matching under the recognition that adaptation for LOO is a small perturbation on the full data posterior. The latter class of methods define transformations based on relaxing various statistical objectives: in our case the variance of the IS estimator and the KL divergence between the transformed distribution and the statistics of the LOO fold. Being model-specific, the gradient flow transformations require evaluating Jacobian determinants. While these quantities are generally readily available through auto-differentiation, we derive closed-form expressions in the case of logistic regression and shallow ReLU activated neural networks. We tested the methodology on an $n\ll p$ dataset that is known to produce unstable LOO IS weights.

Mingtao Xia, Xiangting Li, Qijing Shen et al.

We provide an analysis of the squared Wasserstein-2 ($W_2$) distance between two probability distributions associated with two stochastic differential equations (SDEs). Based on this analysis, we propose the use of a squared $W_2$ distance-based loss functions in the \textit{reconstruction} of SDEs from noisy data. To demonstrate the practicality of our Wasserstein distance-based loss functions, we performed numerical experiments that demonstrate the efficiency of our method in reconstructing SDEs that arise across a number of applications.