Neural Discovery of Memory and Nonlocal Kernels in Integro-Differential Equations with Constrained Kolmogorov--Arnold Networks

What the math gives to ML

The transferable asset is the hard-constrained Bernstein parameterization of positive, monotone-decreasing, convex kernels: it converts qualitative prior knowledge into coefficient inequalities that hold for every network output, rather than relying on penalty tuning. This can become a trainable causal memory head for sequence models, where the learned lag kernel determines how past value vectors are aggregated and is therefore interpretable as a nonparametric replacement for a fixed exponential or power-law decay. The most direct experiment is to add this branch to a small Transformer or SSM, compare it with unconstrained learned lag weights and exponential memory at equal parameter count, and test whether the shape constraints improve extrapolation and stability under sparse or noisy training data.