Parameter-Minimal Neural DE Solvers via Horner Polynomials

TL;DR

This paper introduces a minimal-parameter neural approach using Horner polynomials for solving differential equations, achieving high accuracy with significantly fewer parameters than traditional methods.

Contribution

It presents a novel low-parameter neural architecture based on Horner polynomials and a spline extension for efficient differential equation solving.

Findings

Outperforms small MLP and sinusoidal models on benchmarks

Accurately matches solutions and derivatives with minimal parameters

Demonstrates a practical accuracy-parameter trade-off

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.