KHRONOS: a Kernel-Based Neural Architecture for Rapid, Resource-Efficient Scientific Computation

TL;DR

KHRONOS addresses the curse of dimensionality in scientific computation by unifying model-free, model-based, and inverse learning within a kernel-based surrogate. It represents targets as hierarchical, per-dimension kernel expansions that form separable tensor-product modes, enabling fast, differentiable solutions with strong parameter efficiency. Empirical results on 8-D borehole and 20-D Sobol-G benchmarks, plus a 2D Poisson PDE, show substantial accuracy gains over Kolmogorov-Arnold networks and FEM baselines, with sub-millisecond to microsecond inference and effective batched Gauss-Newton inversion for level-set problems. The framework promises impactful applications in edge computing, online control, and image-based tasks, while highlighting limitations related to regular grids and kernel choices that motivate future extensions to unstructured meshes and higher-order kernels.

Abstract

Contemporary models of high dimensional physical systems are constrained by the curse of dimensionality and a reliance on dense data. We introduce KHRONOS (Kernel Expansion Hierarchy for Reduced Order, Neural Optimized Surrogates), an AI framework for model based, model free and model inversion tasks. KHRONOS constructs continuously differentiable target fields with a hierarchical composition of per-dimension kernel expansions, which are tensorized into modes and then superposed. We evaluate KHRONOS on a canonical 2D, Poisson equation benchmark: across 16 to 512 degrees of freedom (DoFs), it obtained L_2-square errors of 5e-4 down to 6e-11. This represents a greater than 100-fold gain over Kolmogorov Arnold Networks (which itself reports a 100 times improvement on MLPs/PINNs with 100 times fewer parameters) when controlling for the number of parameters. This also represents a 1e6-fold improvement in L_2-square error compared to standard linear FEM at comparable DoFs. Inference complexity is dominated by inner products, yielding sub-millisecond full-field predictions that scale to an arbitrary resolution. For inverse problems, KHRONOS facilitates rapid, iterative level set recovery in only a few forward evaluations, with sub-microsecond per sample latency. KHRONOS's scalability, expressivity, and interpretability open new avenues in constrained edge computing, online control, computer vision, and beyond.

Figures (5)

• Figure 1: Schematic of KHRONOS's architecture. Each input feature $x_p$ is mapped via a kernel expansion (layers 1-$L$) defined on a small number of nodes (yellow) within each segment. Per-dimension feature vectors are projected by learned weights $w$ and then combined via a tensor product ($\prod$) to form each mode. Finally, $M$ such modes are summed ($\sum$) to yield the surrogate.
• Figure 2: Plot of convergence toward perfect accuracy, $1-R^2\rightarrow0$, as trainable parameters increase for each surrogate model
• Figure 3: Exact solution, model prediction and the normalized absolute error for a 16-parameter KHRONOS solve
• Figure 4: Plot showing $L_2^2$ error and $H_1^2$ error as degrees of freedom increase. The left plot additionally shows the $L^2_2$ errors achieved by P1- and P2 Lagrange element FEM. $N^{-6},N^{-5},N^{-4}$ scaling laws for $L_2^2$ and $N^{-4}, N^{-3}$ scaling laws for $H^2_1$ errors are shown for reference
• Figure 5: Batch inversion, 400 points, of KHRONOS on a highly non-convex toy example