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Replica exchange stochastic gradient Langevin dynamics (reSGLD) is an effective sampler for non-convex learning in large-scale datasets. However, the simulation may encounter stagnation issues when the high-temperature chain delves too deeply into the distribution tails. To tackle this issue, we propose reflected reSGLD (r2SGLD): an algorithm tailored for constrained non-convex exploration by utilizing reflection steps within a bounded domain. Theoretically, we observe that reducing the diameter of the domain enhances mixing rates, exhibiting a quadratic behavior. Empirically, we test its performance through extensive experiments, including identifying dynamical systems with physical constraints, simulations of constrained multi-modal distributions, and image classification tasks. The theoretical and empirical findings highlight the crucial role of constrained exploration in improving the simulation efficiency. 

The present study develops a physics-constrained neural network (PCNN) to predict sequential patterns and motions of multiphase flows (MPFs), which includes strong interactions among various fluid phases. To predict the order parameters, which locate individual phases in the future time, a neural network (NN) is applied to quickly infer the dynamics of the phases by encoding observations. The multiphase consistent and conservative boundedness mapping algorithm (MCBOM) is next implemented to correct the predicted order parameters. This enforces the predicted order parameters to strictly satisfy the mass conservation, the summation of the volume fractions of the phases to be unity, the consistency of reduction, and the boundedness of the order parameters. Then, the density of the fluid mixture is updated from the corrected order parameters. Finally, the velocity in the future time is predicted by another NN with the same network structure, but the conservation of momentum is included in the loss function to shrink the parameter space. The proposed PCNN for MPFs sequentially performs (NN)-(MCBOM)-(NN), which avoids nonphysical behaviors of the order parameters, accelerates the convergence, and requires fewer data to make predictions. Numerical experiments demonstrate that the proposed PCNN is capable of predicting MPFs effectively. 

Abstract With the explosive growth of biomarker data in Alzheimer’s disease (AD) clinical trials, numerous mathematical models have been developed to characterize disease-relevant biomarker trajectories over time. While some of these models are purely empiric, others are causal, built upon various hypotheses of AD pathophysiology, a complex and incompletely understood area of research. One of the most challenging problems in computational causal modeling is using a purely data-driven approach to derive the model’s parameters and the mathematical model itself, without any prior hypothesis bias. In this paper, we develop an innovative data-driven modeling approach to build and parameterize a causal model to characterize the trajectories of AD biomarkers. This approach integrates causal model learning, population parameterization, parameter sensitivity analysis, and personalized prediction. By applying this integrated approach to a large multicenter database of AD biomarkers, the Alzheimer’s Disease Neuroimaging Initiative, several causal models for different AD stages are revealed. In addition, personalized models for each subject are calibrated and provide accurate predictions of future cognitive status.