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国家预印本平台全新架构的高精度可训练PDE 算子：AI 参数数量存在唯一性的数学证明与实验验证
High-Precision Trainable PDE Operators with a Novel Architecture: A Mathematical Proof and Experimental Verification of the Uniqueness of AI Parameters
V2
摘要
AI模型普遍存在分布外（OOD）泛化产生幻觉的问题，难以用于严格科学计算。现有物理信息神经网络（PINNs）、傅里叶神经算子（FNOs）等主流方案均为在工程层面的软正则约束提供物理先验，未能突破AI底层的拓扑结构缺陷。
在机器学习的流形假设（Manifold Hypothesis）中：所有的AI模型，本质上就是一种根据输入数据拟合特定高维流形的可训练算子。而这类高维函数逼近与全局收敛性问题的数学基础，早在1943年，由Courant奠定的古典伽辽金有限元变分理论中得到系统性建立与严格证明。
以此为基础，本文融合了有限元与线性代数理论，证明了AI参数数量的存在唯一性：为彻底消除非平凡零空间与 OOD 幻觉，AI 参数数量必须严格等于AI训练集下 Galerkin 投影的非线性基底数，即 $N_{AI}=\dim(V_h)=N_{basis}$(其中 $V_h$ 是 Galerkin 投影的有限维子空间，$N_{\text{basis}}$是该空间内的非线性基底数)。
数值实验结果表明，全新架构仅需$O(1)$的可训练参数与简单的ADAM梯度下降方法，即可实现 FP64 双精度浮点极限精度的零幻觉 OOD 泛化：$MSELoss=O(10^{-32})$，彻底解决了OOD泛化问题。
Abstract
Artificial intelligence (AI) models are generally prone to hallucinations arising from out-of-distribution (OOD) generalization, rendering them unsuitable for rigorous scientific computation. Existing mainstream approaches, such as Physics-Informed Neural Networks (PINNs) and Fourier Neural Operators (FNOs), merely provide physical priors through soft regularization constraints at the engineering level, failing to address the fundamental topological defects inherent in AI architectures. Within the manifold hypothesis of machine learning: all AI models are essentially trainable operators that fit specific high-dimensional manifolds based on input data. The mathematical foundation for such high-dimensional function approximation and global convergence issues was systematically established and rigorously proven as early as 1943 through the classical Galerkin finite element variational theory pioneered by Courant. Building upon this foundation, this paper integrates finite element theory with linear algebra to prove the existence and uniqueness of AI parameters: to completely eliminate non-trivial null spaces and OOD hallucinations, the number of AI parameters must strictly equal the number of nonlinear basis functions in the Galerkin projection under the AI training set, i.e., $N_{AI}=\dim(V_h)=N_{basis}$ (where $V_h$ denotes the finite-dimensional subspace of the Galerkin projection, and $N_{\text{basis}}$ represents the number of nonlinear basis functions within this subspace). Numerical experimental results demonstrate that the novel architecture achieves zero-hallucination OOD generalization at FP64 double-precision floating-point limit accuracy using only $O(1)$ trainable parameters and simple ADAM gradient descent: $MSELoss=O(10^{-32})$, thereby completely resolving the OOD generalization problem.关键词
计算数学/有限元/伽辽金投影/等参变换/秩-零化度定理Key words
Math.NA/FEM/Galerkin projection/Isoparametric Transformation/Rank-Nullity Theorem引用本文复制引用
杜秋实.全新架构的高精度可训练PDE 算子：AI 参数数量存在唯一性的数学证明与实验验证[EB/OL].(2026-04-17)[2026-04-28].https://sinoxiv.napstic.cn/article/25792260.学科分类
数学/计算技术、计算机技术/物理学版本历史
| 版本 | 时间 | DOI号 | 操作 |
|---|---|---|---|
| v2（当前版本） | 2026-04-28 09:10:01 | 10.12383/202603300001V2 | 全文下载 |
| v1 | 2026-04-17 16:20:47 | 10.12383/202603300001V1 | 全文下载 |
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