Parameter-Minimal Neural DE Solvers via Horner Polynomials
This work addresses resource-efficient scientific modeling for differential equations, though it is incremental as it builds on existing polynomial and piecewise methods.
The authors tackled the problem of solving differential equations with minimal parameters by proposing a neural architecture based on Horner-factorized polynomials, which achieved accurate solutions and derivatives using only tens of parameters, outperforming small MLP and sinusoidal baselines in benchmarks.
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.