证明了神经算子在形状变化下的误差边界，支持跨形变的泛化能力。
Neural Shape Operator Surrogates -- Expression Rate Bounds
- 通过参数化形变将方程映射到参考域，构建解析型参数化问题
- 给出神经与谱算子代理的误差界和收敛速率，统一适用于可微形变族
- 为神经算子处理椭圆/抛物型方程及边界积分方程提供理论支撑
我们针对一类与参考域 $D_{ref}$ 微分同胚的域族上偏微分方程与边界积分方程的解算子，建立了算子代理的误差界。通过仿射参数化形状编码将原方程拉回参考域 $D_{ref}$，得到一系列在 $D_{ref}$ 上的解析参数化方程。给出了保证（关于参数一致）适定性的充分条件，意味着参数解族的存在性、唯一性与稳定性。通过回顾一系列椭圆与抛物型方程的解析性结果，验证了抽象假设。量化参数解析性意味着存在有限参数离散逼近，其收敛率与参数数量 $N$ 相关。我们构造性地证明了神经算子与谱算子代理对形状-解映射的存在性，并给出统一的误差界与收敛率保证。允许主成分形状编码与框架解码器。结果支持神经算子在椭圆与抛物型方程及边界积分方程中实现数据到解映射并跨参数形状族泛化的经验观察。
原文摘要 · Abstract (English)
We prove error bounds for operator surrogates of solution operators for partial differential and boundary integral equations on families of domains which are diffeomorphic to one common reference (or latent) domain $D_{ref}$. The pullback of the PDE to $D_{ref}$ via affine-parametric shape encoding produces a collection of holomorphic parametric PDEs on $D_{ref}$. Sufficient conditions for (uniformly with respect to the parameter) well-posedness are given, implying existence, uniqueness and stability of parametric solution families on $D_{ref}$. We illustrate the abstract hypotheses by reviewing recent holomorphy results for a suite of elliptic and parabolic PDEs. Quantified parametric holomorphy implies existence of finite-parametric, discrete approximations of the parametric solution families with convergence rates in terms of the number $N$ of parameters. We obtain constructive proofs of existence of Neural and Spectral Operator surrogates for the shape-to-solution maps with error bounds and convergence rate guarantees uniform on the collection of admissible shapes. We admit principal-component shape encoders and frame decoders. Our results support in particular the (empirically reported) ability of neural operators to realize data-to-solution maps for elliptic and parabolic PDEs and BIEs that generalize across parametric families of shapes.
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