NeSTR：高斯SPDE的神经S-变换重建
NeSTR: Neural S-Transform Reconstruction for Gaussian SPDEs
- KTH Royal Institute of Technology（皇家理工学院）
- Digital Futures（数字未来中心）
- BI Norwegian Business School（挪威商业管理学院）
- University of Ljubljana（卢布尔雅那大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
NeSTR通过有限维子空间上的S-变换学习高斯SPDE的解，利用逆S-变换重建原理和有限模式学习，将神经逼近与Wiener混沌展开关联，数值测试验证了其稳定恢复高阶混沌系数的能力。
AI中文摘要：
我们提出了用于高斯随机偏微分方程（SPDEs）的神经S-变换重建（NeSTR）。学习对象不是随机解本身，而是其在白噪声测试空间的有限维子空间上的S-变换。对于Wick型SPDEs，NeSTR为受限S-变换产生一个确定性参数PDE，随机解从该变换在原点的泰勒系数中恢复。因此，神经逼近直接与解的Wiener混沌展开相关联。NeSTR的主要新颖之处在于将逆S-变换重建原理与有限模式学习相结合：我们证明了恢复的混沌系数的一致性，分离了噪声模式、混沌截断、确定性求解器和神经逼近误差，并为受限变换给出了确定性学习公式。对加性随机热方程和乘性Wick热方程的数值测试表明，变换变量中的多项式谱表示为高阶混沌系数提供了稳定的访问。所得重建将依赖于方程的确定性系数场与可复用的高斯-埃尔米特基分离。因此，NeSTR学习一个解析生成函数，其导数恢复Wiener混沌系数，而不是直接学习随机轨迹。
英文摘要：
We introduce Neural $S$-Transform Reconstruction (NeSTR) for Gaussian stochastic partial differential equations (SPDEs). The learned object is not the random solution itself, but its $S$-Transform on a finite dimensional subspace of the white noise test space. For Wick type SPDEs, NeSTR yields a deterministic parametric PDE for the restricted $S$-Transform, and the stochastic solution is recovered from the Taylor coefficients of this transform at the origin. Thus the neural approximation is tied directly to the Wiener chaos expansion of the solution. The main novelty of NeSTR is an inverse $S$-Transform reconstruction principle combined with finite mode learning: we prove consistency of the recovered chaos coefficients, separate noise mode, chaos truncation, deterministic solver, and neural approximation errors, and give a deterministic learning formulation for the restricted transform. Numerical tests on an additive stochastic heat equation and a multiplicative Wick heat equation show that polynomial spectral representations in the transform variables provide stable access to higher order chaos coefficients. The resulting reconstruction separates the equation dependent deterministic coefficient fields from a reusable Gaussian--Hermite basis. Consequently, NeSTR learns an analytic generating function whose derivatives recover the Wiener chaos coefficients, rather than learning stochastic trajectories directly.
