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.

翻译：我们提出了一种参数最小化的神经架构，通过将假设类限制为霍纳分解多项式来求解微分方程，从而得到仅包含少量可学习系数的隐式可微分试验解。通过固定低阶多项式自由度，初始条件在构造时被精确满足，因此训练仅专注于在配置点上匹配微分方程残差。为了在不放弃低参数体系的前提下减少近似误差，我们引入了一种分段（"类样条"）扩展方法，该方法在子区间上训练多个小型霍纳模型，同时在分段边界处强制连续性（及一阶导数连续性）。在示例性常微分方程基准测试和热方程案例中，仅包含数十（或更少）参数的霍纳网络能够精确匹配解及其导数，并在相同训练设置下优于小型多层感知机和正弦表示基线，这为资源高效的科学建模展示了实用的精度-参数权衡关系。