Ramansh Sharma

We present a local and sparse kernel method for learning the solution operators of incompressible flows. It builds on our earlier property-preserving kernel method, whose global output reconstruction and dense operator regression scale poorly. Our method replaces the global output reconstruction by local divergence-free interpolants with divergence-free polynomials of any degree, solved in parallel on the GPU. These interpolants are divergence free, periodic on periodic domains, and, with a multiscale kernel, approach Kolmogorov statistics under refinement. A compactly supported kernel, positive semidefinite in any input dimension, makes the operator regression sparse.

We introduce a partition-of-unity mixture-of-experts framework for localized operator learning, using geometry-aware gating networks to blend local experts through smooth spatial partitions. Our HiRefPOU architecture organizes DeepONet representations into nested parent-child partitions with residual refinement while preserving global continuity. We also incorporate spatial adaptivity into Fourier Neural Operators. On heterogeneous Darcy and reaction-diffusion benchmarks, HiRefPOU improves accuracy over global DeepONet and static POU-MoE baselines, with interpretable partitions that align with interfaces and regions of rapid solution variation.

We develop a lossy compression method for incompressible velocity fields that analytically preserves zero divergence and, on periodic domains, periodicity at every compression tolerance. The method represents velocities using property-preserving matrix-valued kernels and compresses their coefficients with standard lossy compressors. We develop low-rank, multilevel, and redundant kernel representations, derive bounds linking coefficient compression errors to velocity errors, and evaluate accuracy and storage on cavity, Taylor-Green, and species-transport flows.

We present a novel property-preserving kernel-based operator learning method for incompressible flows governed by the incompressible Navier-Stokes equations. Operator learning offers efficient surrogate models, but current neural operators fail to exactly enforce physical properties such as incompressibility, periodicity, and turbulence. Our method maps input functions to expansion coefficients of output functions in a property-preserving kernel basis, ensuring that predicted velocity fields analytically and simultaneously preserve the aforementioned physical properties. We evaluate the method on challenging 2D and 3D, laminar and turbulent, incompressible flow problems. Our method achieves up to six orders of magnitude lower relative errors upon generalization and trains up to five orders of magnitude faster compared to neural operators.

We present a novel deep operator network (DeepONet) architecture for operator learning, the ensemble DeepONet, that allows for enriching the trunk network of a single DeepONet with multiple distinct trunk networks. This trunk enrichment allows for greater expressivity and generalization capabilities over a range of operator learning problems. We also present a spatial mixture-of-experts (MoE) DeepONet trunk network architecture that utilizes a partition-of-unity (PoU) approximation to promote spatial locality and model sparsity in the operator learning problem.

We present a new technique for the accelerated training of physics-informed neural networks (PINNs): discretely-trained PINNs (DT-PINNs). DT-PINNs are trained by replacing exact spatial derivatives with high-order accurate numerical discretizations computed using meshless radial basis function-finite differences (RBF-FD) and applied via sparse-matrix vector multiplication. Additionally, though traditional PINNs (vanilla-PINNs) are typically stored and trained in 32-bit floating-point (fp32) on the GPU, we show that for DT-PINNs, using fp64 on the GPU leads to significantly faster training times than fp32 vanilla-PINNs with comparable accuracy. Our results show that fp64 DT-PINNs offer a superior cost-accuracy profile to fp32 vanilla-PINNs.

We propose a Knowledge Management Framework based on NLP techniques that would tackle challenges such as resource-intensive nature, lack of comprehensive information sources, and overlapping jurisdictions in policy analysis. The framework is designed to be platform-, language- and policy-agnostic. To classify financial incentives in restoration policies, both Sentence-BERT and Cross-Encoders performed well. For sentence classification inference with Sentence-BERT, a random forest classifier can be used to assign a category to a given sentence using the Sentence-BERT learned embeddings.