Harish S. Bhat

Harish S. Bhat, R. W. M. A. Madushani

We develop and analyze a method, density tracking by quadrature (DTQ), to compute the probability density function of the solution of a stochastic differential equation. The derivation of the method begins with the discretization in time of the stochastic differential equation, resulting in a discrete-time Markov chain with continuous state space. At each time step, DTQ applies quadrature to solve the Chapman-Kolmogorov equation for this Markov chain. In this paper, we focus on a particular case of the DTQ method that arises from applying the Euler-Maruyama method in time and the trapezoidal quadrature rule in space. Our main result establishes that the density computed by DTQ converges in $L^1$ to both the exact density of the Markov chain (with exponential convergence rate), and to the exact density of the stochastic differential equation (with first-order convergence rate). We establish a Chernoff bound that implies convergence of a domain-truncated version of DTQ. We carry out numerical tests to show that the empirical performance of DTQ matches theoretical results, and also to demonstrate that DTQ can compute densities several times faster than a Fokker-Planck solver, for the same level of error.

Hardeep Bassi, Yuanran Zhu, Erika Ye et al.

We present RELift (Restrict, Evolve, Lift), a two-phase learning framework that couples coarse-grid numerical solvers with neural operators to super-resolve and forecast fine-grid dynamics for time-dependent partial differential equations (PDEs). In Phase 1, RELift learns a super-resolution operator that maps the solution on a coarse grid to a fine grid. In Phase 2, this learned operator is composed with a coarse-grid numerical integrator to construct an effective fine-grid propagator for the governing equation. We benchmark RELift on three canonical two-dimensional PDEs of increasing dynamical complexity -- the heat equation, the wave equation, and the incompressible Navier--Stokes equations -- and we further demonstrate its performance on a kinetic electron dynamics case study via the 1D1V Vlasov--Poisson system. Across all examples, RELift delivers high-fidelity super-resolution (Phase 1) and accurate long-horizon rollouts (Phase 2), outperforming standard super-resolution and neural operator baselines in both field-level error metrics and physics-relevant diagnostics. Finally, we provide error analysis of the effective fine-grid propagator, characterizing how approximation errors accumulate over time and explaining the observed numerical stability of the RELift framework.

Harish S. Bhat, Majerle E. Reeves, Sidra Goldman-Mellor

When faced with severely imbalanced binary classification problems, we often train models on bootstrapped data in which the number of instances of each class occur in a more favorable ratio, e.g., one. We view algorithmic inequity through the lens of imbalanced classification: in order to balance the performance of a classifier across groups, we can bootstrap to achieve training sets that are balanced with respect to both labels and group identity. For an example problem with severe class imbalance---prediction of suicide death from administrative patient records---we illustrate how an equity-directed bootstrap can bring test set sensitivities and specificities much closer to satisfying the equal odds criterion. In the context of naïve Bayes and logistic regression, we analyze the equity-directed bootstrap, demonstrating that it works by bringing odds ratios close to one, and linking it to methods involving intercept adjustment, thresholding, and weighting.

Harish S. Bhat

We consider the problem of learning an interpretable potential energy function from a Hamiltonian system's trajectories. We address this problem for classical, separable Hamiltonian systems. Our approach first constructs a neural network model of the potential and then applies an equation discovery technique to extract from the neural potential a closed-form algebraic expression. We demonstrate this approach for several systems, including oscillators, a central force problem, and a problem of two charged particles in a classical Coulomb potential. Through these test problems, we show close agreement between learned neural potentials, the interpreted potentials we obtain after training, and the ground truth. In particular, for the central force problem, we show that our approach learns the correct effective potential, a reduced-order model of the system.