基于逆玻恩级数的神经算子
Inverse Born series based neural operators
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中文总结 AI 辅助
该研究提出将逆玻恩级数与神经算子结合的框架，构造神经算子近似逆玻恩级数的算子展开，在逆散射和Calderón问题的数值实验中表现优异，为学习非线性逆算子提供了有效途径。
中文摘要 AI 辅助
逆玻恩级数为表示无穷维函数空间间的非线性逆映射提供了通用的微扰框架，已在偏微分方程和积分方程控制的逆问题中得到大量应用。受其算子理论结构的启发，我们开发了一套系统框架用于构造神经算子，以近似逆玻恩级数中出现的算子展开式。我们的方法将逆玻恩级数的解析表示与神经算子的表达能力相结合，在保留底层算子结构的同时，实现了非线性逆映射的数据驱动近似。该框架适用于广泛的逆问题，且在一般泛函分析框架下提出。为验证其实际性能，我们选取两个代表性示例：逆散射问题和Calderón（电阻抗断层扫描）问题。数值实验表明，所构造的神经算子可精确近似逆玻恩展开式，并在一系列测试案例中生成高质量的重建结果。这些结果表明，所提方法为学习非线性逆算子提供了一种有效且计算高效的途径，也为将经典算子展开与现代神经算子架构相结合指明了有前景的方向。
英文摘要
The inverse Born series provides a general perturbative framework for representing nonlinear inverse maps between infinite-dimensional function spaces and has found numerous applications in inverse problems governed by partial differential equations and integral equations. Motivated by its operator-theoretic structure, we develop a systematic framework for constructing neural operators that approximate the operator expansions arising in the inverse Born series. Our approach combines the analytical representation of the inverse Born series with the expressive power of neural operators, yielding data-driven approximations of the nonlinear inverse map while preserving the underlying operator structure. The proposed framework is applicable to a broad class of inverse problems and is presented in a general functional-analytic setting. To demonstrate its practical performance, we consider two representative examples: inverse scattering and the Calderón (electrical impedance tomography) problem. Numerical experiments show that the constructed neural operators accurately approximate the inverse Born expansions and produce high-quality reconstructions across a range of test cases. These results indicate that the proposed methodology provides an effective and computationally efficient approach for learning nonlinear inverse operators and suggests a promising direction for integrating classical operator expansions with modern neural operator architectures.
