Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives, which commonly require computationally-demanding numerical differentiation such as perturbation techniques. In particular, Gauss-Newton methods are used for nonlinear inverse problems that require iterative updates to be computed from the Jacobian and allow for flexible incorporation of prior knowledge. Computationally more efficient alternatives are Quasi-Newton methods, where the repeated computation of the Jacobian is replaced by an approximate update, but unfortunately are often too restrictive for highly ill-posed problems. To overcome this limitation, we present a highly efficient data-driven Quasi-Newton method applicable to nonlinear inverse problems, by using the singular value decomposition and learning a mapping from model outputs to the singular values to compute the updated Jacobian. Enabling time-critical applications and allowing for flexible incorporation of prior knowledge necessary to solve ill-posed problems. We present results for the highly non-linear inverse problem of electrical impedance tomography with experimental data.

中文翻译：

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基于学习的奇异值的非线性反问题的一种有效的拟牛顿法

解决工程和物理科学中的复杂优化问题，需要对多维函数导数进行重复计算，而多维函数导数通常需要计算要求的数值微分，例如微扰技术。特别是，高斯-牛顿法用于非线性反问题，这些反问题需要从雅可比行列式中计算出迭代更新，并允许灵活地合并先验知识。在计算上更有效的替代方法是拟牛顿法，其中用近似更新代替了对雅可比行列式的重复计算，但不幸的是，对于高度不适的问题，它们往往过于严格。为了克服这一限制，我们提出了一种适用于非线性逆问题的高效数据驱动的拟牛顿法，通过使用奇异值分解并学习从模型输出到奇异值的映射来计算更新的雅可比行列式。支持时间紧迫的应用程序，并允许灵活地合并解决不适当地问题所需的先验知识。我们用实验数据给出了电阻抗层析成像的高度非线性反问题的结果。