用霍纳多项式构建极简神经微分方程求解器，参数少且精度高。
Parameter-Minimal Neural DE Solvers via Horner Polynomials
- 以霍纳多项式约束模型假设空间，仅需少量可学习系数。
- 在张量型测试中，十参数以内模型即准确匹配解与导数。
- 适合资源受限场景下的科学建模，对小模型更友好。
我们提出一种参数极简的神经微分方程求解架构，通过将假设空间限制为霍纳因子分解多项式，得到一个隐式、可微的试解函数，仅需少量可学习系数。初始条件通过固定低阶多项式的自由度精确满足，训练仅聚焦于拟合配点处的微分方程残差。为在不牺牲低参数特性的情况下降低近似误差，我们引入分段（类样条）扩展，即在子区间上训练多个小型霍纳模型，并在区间边界强制保证连续性及一阶导数连续性。在典型常微分方程基准和热方程示例上，仅含数十个（或更少）参数的霍纳网络能准确逼近解及其导数，在相同训练条件下优于小型MLP和正弦表示基线，展现出面向资源高效科学建模的实际精度-参数权衡。
原文摘要 · Abstract (English)
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.
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