arXiv subjects

Satish Chandran

Publications and source records attributed to Satish Chandran.

3 recordsLinked to original sources

arXiv2026

Structure-Preserving Detailed-Balance Master-Equation Discretizations for Fokker--Planck Equations

We develop a variational--Markov construction of detailed-balance master-equation discretizations for Fokker--Planck equations directly from their energy--dissipation laws. Rather than discretizing the differential operator, we represent mass transfer between neighboring grid points by two directional jump rates. On each edge, the discrete energy--dissipation law determines the net flux, while detailed balance fixes the rate ratio; a local reconstruction of edge quantities from neighboring grid-point values then uniquely determines both rates. A logarithmic-mean reconstruction recovers the classical Scharfetter--Gummel/Wang--Peskin--Elston rates, while alternative reconstructions yield other reversible schemes. The resulting semi-discrete systems conserve mass, preserve nonnegativity, satisfy detailed balance, and dissipate a discrete free energy. The same edgewise construction extends to state-dependent and degenerate mobilities, nonlocal interaction energies and higher-dimensional problems. The corresponding implicit and semi-implicit fully discrete schemes are linear, mass-conservative, positivity-preserving, and energy-stable, with a time-step restriction required only for nonlocal interaction energies whose kernel is not negative semidefinite. Numerical experiments confirm the expected spatial convergence rates of all detailed-balance schemes and verify their equilibrium accuracy, positivity preservation, and free-energy decay with discontinuous potentials, saturation, nonlocal interactions, and two-dimensional problems.

math.NA↗

arXiv2026

Spectral Regularization for Diffusion Models

Diffusion models are typically trained using pointwise reconstruction objectives that are agnostic to the spectral and multi-scale structure of natural signals. We propose a loss-level spectral regularization framework that augments standard diffusion training with differentiable Fourier- and wavelet-domain losses, without modifying the diffusion process, model architecture, or sampling procedure. The proposed regularizers act as soft inductive biases that encourage appropriate frequency balance and coherent multi-scale structure in generated samples. Our approach is compatible with DDPM, DDIM, and EDM formulations and introduces negligible computational overhead. Experiments on image and audio generation demonstrate consistent improvements in sample quality, with the largest gains observed on higher-resolution, unconditional datasets where fine-scale structure is most challenging to model.

cs.LG↗

arXiv2026

NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equations

We introduce NewPINNs, a physics-informing learning framework that couples neural networks with conventional numerical solvers for solving differential equations. Rather than enforcing governing equations and boundary conditions through residual-based loss terms, NewPINNs integrates the solver directly into the training loop and defines learning objectives through solver-consistency. The neural network produces candidate solution states that are advanced by the numerical solver, and training minimizes the discrepancy between the network prediction and the solver-evolved state. This pull-push interaction enables the network to learn physically admissible solutions through repeated exposure to the solver's action, without requiring problem-specific loss engineering or explicit evaluation of differential equation residuals. By delegating the enforcement of physics, boundary conditions, and numerical stability to established numerical solvers, NewPINNs mitigates several well-known failure modes of standard physics-informed neural networks, including optimization pathologies, sensitivity to loss weighting, and poor performance in stiff or nonlinear regimes. We demonstrate the effectiveness of the proposed approach across multiple forward and inverse problems involving finite volume, finite element, and spectral solvers.

cs.LG↗