🤖 AI Summary
This work proposes a minimal neural architecture based on Horner-factorized polynomials to address the challenge of large parameter counts in neural networks for solving differential equations, which hinders deployment in resource-constrained settings. The method constructs an implicit, differentiable trial solution with only a few learnable coefficients that inherently satisfies initial conditions. By integrating a piecewise-continuous (spline-like) stitching strategy with a residual collocation scheme, the approach achieves high accuracy and smoothness in both the solution and its derivatives while drastically reducing model size. Experiments on ordinary differential equations and the heat equation demonstrate that the proposed method accurately approximates the true solutions and their derivatives using merely dozens of parameters, outperforming comparably sized small MLPs and sinusoidal representation networks.
📝 Abstract
We propose a parameter-minimal neural architecture for solving differential equations by restricting the hypothesis class to Horner-factorized polynomials, yielding an implicit, differentiable trial solution with only a small set of learnable coefficients. Initial conditions are enforced exactly by construction by fixing the low-order polynomial degrees of freedom, so training focuses solely on matching the differential-equation residual at collocation points. To reduce approximation error without abandoning the low-parameter regime, we introduce a piecewise ("spline-like") extension that trains multiple small Horner models on subintervals while enforcing continuity (and first-derivative continuity) at segment boundaries. On illustrative ODE benchmarks and a heat-equation example, Horner networks with tens (or fewer) parameters accurately match the solution and its derivatives and outperform small MLP and sinusoidal-representation baselines under the same training settings, demonstrating a practical accuracy-parameter trade-off for resource-efficient scientific modeling.