AI 中文总结
研究构建由固定神经元浅层ReLU$^k$神经网络生成的有限维德拉姆子复形，引入神经微分形式空间，在特定假设下证明其正合性并给出线性独立条件，数值实验验证了稳定离散化及收敛率等。
AI 中文摘要
我们构建了由固定神经元浅层ReLU$^k$神经网络生成的有限维德拉姆子复形，这类空间已知能提供最优逼近率。对于$s_i(x)=\omega_i\cdot x+b_i$形式的神经元，我们引入神经微分形式空间：其系数为ReLU$^k$脊函数$\sigma_{k - p}(s_i)$的微分$p$形式。这些空间与外微分兼容，因对ReLU幂求导会使其阶数降低一阶，且对每个固定神经元，求导相当于与固定的1形式$ds_i$进行外积。在最低阶族$\{\sigma_{k - d}(s_i)\}_{i = 1}^n$的线性独立假设下，全局复形分解为独立的神经元级科祖尔复形。我们证明了在任意维度下的正合性，并为所需的线性独立提供了几何充分条件。基于所得复形的数值实验证明了稳定离散化以及与基础逼近理论一致的收敛率，且在所考虑的特征值问题中未出现虚假模式。
英文摘要
We construct finite-dimensional de Rham subcomplexes generated by fixed-neuron shallow ReLU$^k$ neural networks, a class of spaces known to provide optimal approximation rates. For neurons of the form $s_i(x)=ω_i\cdot x+b_i$, we introduce spaces of neural differential forms: differential $p$-forms whose coefficients are the ReLU$^k$ ridge functions $σ_{k-p}(s_i)$. These spaces are compatible with the exterior derivative because differentiating a ReLU power lowers its order by one, and for each fixed neuron, differentiation amounts to exterior multiplication by the fixed one-form $ d s_i$. Under a linear independence assumption on the lowest-order family $\{σ_{k-d}(s_i)\}_{i=1}^n$, the global complex decomposes into independent neuron-wise Koszul complexes. We prove exactness in arbitrary dimension and provide a geometric sufficient condition for the required linear independence. Numerical experiments based on the resulting complex provide evidence of stable discretizations and of convergence rates consistent with the underlying approximation theory, and exhibit no spurious modes in eigenvalue problems considered.
