用数学物理先验让神经网络精确求解偏微分方程
The Vekua Layer: Exact Physical Priors for Implicit Neural Representations via Generalized Analytic Functions
- 基于广义解析函数理论设计可微谱层，将学习转为凸优化
- 重建误差低至10⁻³³，噪声下仍保持稳定（MSE≈0.03）
- 可从局部边界数据外推全局场，具物理可解释性
隐式神经表征（INRs）在参数化物理场方面表现出强大能力，但常受频谱偏差和非凸优化计算成本影响。本文提出Vekua层（VL），一种基于广义解析函数经典理论的可微谱方法。通过将假设空间限制在控制微分算子的核内（使用调和与傅里叶-贝塞尔基），VL将学习任务从迭代梯度下降转化为严格凸的最小二乘问题，通过线性投影求解。我们在同质椭圆型偏微分方程上对比了SIRENs。结果表明，VL在精确重构任务中达到机器精度（MSE≈10⁻³³），在不相干传感器噪声下表现更优（MSE≈0.03），有效充当物理信息谱滤波器。此外，我们证明了VL可通过解析延拓实现从部分边界数据到全局场的‘全息’外推，这是标准坐标基近似所不具备的能力。
原文摘要 · Abstract (English)
Implicit Neural Representations (INRs) have emerged as a powerful paradigm for parameterizing physical fields, yet they often suffer from spectral bias and the computational expense of non-convex optimization. We introduce the Vekua Layer (VL), a differentiable spectral method grounded in the classical theory of Generalized Analytic Functions. By restricting the hypothesis space to the kernel of the governing differential operator -- specifically utilizing Harmonic and Fourier-Bessel bases -- the VL transforms the learning task from iterative gradient descent to a strictly convex least-squares problem solved via linear projection. We evaluate the VL against Sinusoidal Representation Networks (SIRENs) on homogeneous elliptic Partial Differential Equations (PDEs). Our results demonstrate that the VL achieves machine precision ($\text{MSE} \approx 10^{-33}$) on exact reconstruction tasks and exhibits superior stability in the presence of incoherent sensor noise ($\text{MSE} \approx 0.03$), effectively acting as a physics-informed spectral filter. Furthermore, we show that the VL enables "holographic" extrapolation of global fields from partial boundary data via analytic continuation, a capability absent in standard coordinate-based approximations.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。