用切比雪夫基替换坐标输入,提升DeepONet对复杂解的建模能力。
Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation
- 用固定切比雪夫谱字典替代原始坐标,引入非周期性先验偏置
- 在多个方程上相对误差降低最高达54%,尤其擅长捕捉尖锐梯度
- 无需额外参数,适合科学计算中非周期边界问题的高效代理建模
Deep Operator Networks (DeepONets) 作为数据驱动算子学习的强大框架,为偏微分方程中的非线性映射提供了灵活的代理模型。然而,标准的主干网络通过全连接层直接处理原始空间或时空坐标,在有界域上难以表示尖锐梯度、边界层等非周期解结构。为此,本文提出谱嵌入的DeepONet(SEDONet),其主干网络采用固定的切比雪夫谱字典而非坐标输入。这种非周期谱嵌入为有界域提供了合理的归纳偏置,使学习到的算子能更准确捕捉细粒度特征,而这些特征对基于傅里叶或仅含MLP的主干网络难以表示。SEDONet在二维泊松方程、一维伯格斯方程、一维对流-扩散方程、阿伦-卡恩方程、洛伦兹-96混沌系统和达西流问题上进行了评估,覆盖椭圆、双曲、抛物、混沌及多尺度问题。在所有基准测试中,SEDONet的相对 $L^2$ 误差均低于或与DeepONet、FEDONet相当,相较于基线DeepONet最高降低54%,在非周期有界问题上持续优于傅里叶嵌入变体。能量谱分析进一步表明,SEDONet更准确地保留了中高频解结构。该框架为DeepONet提供了一种简单、参数无增的改进方式,为科学计算中非线性算子的稳健高效代理建模提供了一种谱方法。
原文摘要 · Abstract (English)
Deep Operator Networks (DeepONets) have emerged as a powerful framework for data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk network, which operates directly on raw spatial or spatiotemporal coordinates through fully connected layers, often struggles to represent sharp gradients, boundary layers, and other non-periodic solution structures on bounded domains. To address these limitations, we introduce the Spectral-Embedded Deep Operator Network (SEDONet), a novel DeepONet architecture in which the trunk is driven by a fixed Chebyshev spectral dictionary instead of coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias for bounded domains, enabling the learned operator to capture fine-scale features that are difficult for Fourier-based or MLP-only trunks to represent. SEDONet is evaluated on the 2-D Poisson equation, 1-D Burgers' equation, 1-D advection-diffusion equation, Allen-Cahn equation, Lorenz-96 chaotic system, and Darcy flow, covering elliptic, hyperbolic, parabolic, chaotic, and multiscale problems. Across all benchmarks, SEDONet consistently achieves the lowest or statistically comparable relative $L^2$ errors among DeepONet, FEDONet, and SEDONet, with improvements of up to 54% over the baseline DeepONet and consistent gains over Fourier-embedded variants on bounded, non-periodic problems. Energy spectrum analyses further demonstrate that SEDONet more accurately preserves intermediate- and high-frequency solution structures. The proposed framework provides a simple, parameter-neutral modification to DeepONets, offering a robust and computationally efficient spectral approach for surrogate modeling of nonlinear operators in scientific computing.
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