提出可保质量的多尺度神经解码器，解决流形学习中的病态逆问题。
RANDSMAPs: Random-Feature/multi-Scale Neural Decoders with Mass Preservation
- 基于随机傅里叶特征与多尺度结构，显式满足守恒律约束
- 在三个基准数据集上实现高精度重建，计算成本低且质量守恒达单机精度
- 适用于含质量守恒需求的物理建模与图像重构场景
我们提出RANDSMAPs（随机特征/多尺度神经解码器，具质量保持性），一种受数值分析启发、可解释的神经解码器，用于在流形学习中求解具有挑战性的病态逆问题，并显式遵守守恒定律。我们证明了标准随机傅里叶特征神经网络在确定性极限下等价于径向基函数插值和双扩散映射（基于几何谐波）解码器。随后建立了RANDSMAP的理论基础，并引入其多尺度变体以捕捉跨尺度结构。我们推导出相应约束优化问题的闭式解，并证明了质量保持性质。数值实验在三个基准问题/数据集上验证：由Lighthill-Whitham-Richards交通流偏微分方程产生的激波结构、2D旋转磁共振脑图像，以及Hughes人群动力学偏微分方程。结果表明，RANDSMAP在低计算成本下实现高重建精度，且在单机精度下维持质量守恒。其原始形式仍适用于经典逆问题，即不施加质量守恒约束的情形。
原文摘要 · Abstract (English)
We introduce RANDSMAPs (Random-feature/multi-scale neural decoders with Mass Preservation), numerical analysis-informed, explainable neural decoders designed to explicitly respect conservation laws when solving the challenging ill-posed pre-image problem in manifold learning. We start by proving the equivalence of vanilla random Fourier feature neural networks to Radial Basis Function interpolation and the double Diffusion Maps (based on Geometric Harmonics) decoders in the deterministic limit. We then establish the theoretical foundations for RANDSMAP and introduce its multiscale variant to capture structures across multiple scales. We formulate and derive the closed-form solution of the corresponding constrained optimization problem and prove the mass preservation property. Numerically, we assess the performance of RANDSMAP on three benchmark problems/datasets with mass preservation obtained by the Lighthill-Whitham-Richards traffic flow PDE with shock waves, 2D rotated MRI brain images, and the Hughes crowd dynamics PDEs. We demonstrate that RANDSMAPs yield high reconstruction accuracy at low computational cost and maintain mass conservation at single-machine precision. In its vanilla formulation, the scheme remains applicable to the classical pre-image problem, i.e., when mass-preservation constraints are not imposed.
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