无需梯度的高斯混合变分推断，高效求解复杂反问题后验分布。
Stable Derivative Free Gaussian Mixture Variational Inference for Bayesian Inverse Problems
- 结合自然梯度与特殊积分规则，实现无梯度更新的高斯混合变分推断。
- 在100维空间中成功处理多峰、无限多峰及弯曲模式分布。
- 适用于前向模型昂贵、梯度不可得的科学计算反问题，如流体初始状态恢复。
本文研究了仅知归一化常数的概率分布近似问题，聚焦于科学计算中大规模反问题的贝叶斯推断。核心挑战包括前向模型重复计算成本高、后验分布多模态以及前向模型梯度不可获取。为此，我们提出一种变分推断框架，结合Fisher-Rao自然梯度与专用积分规则，实现对高斯混合变分族的无梯度更新。所提方法称为无梯度高斯混合变分推断（DF-GMVI），保证协方差正定性与仿射不变性，为复杂后验分布的近似提供稳定高效框架。数值实验验证了其在多重模式、无穷多模式及高维（最高100维）弯曲模式分布中的有效性。实际应用中，该方法成功从正时间点的解数据中恢复纳维-斯托克斯方程的初始条件。
原文摘要 · Abstract (English)
This paper is concerned with the approximation of probability distributions known up to normalization constants, with a focus on Bayesian inference for large-scale inverse problems in scientific computing. In this context, key challenges include costly repeated evaluations of forward models, multimodality, and inaccessible gradients for the forward model. To address them, we develop a variational inference framework that combines Fisher-Rao natural gradient with specialized quadrature rules to enable derivative free updates of Gaussian mixture variational families. The resulting method, termed Derivative Free Gaussian Mixture Variational Inference (DF-GMVI), guarantees covariance positivity and affine invariance, offering a stable and efficient framework for approximating complex posterior distributions. The effectiveness of DF-GMVI is demonstrated through numerical experiments on challenging scenarios, including distributions with multiple modes, infinitely many modes, and curved modes in spaces with up to 100 dimensions. The method's practicality is further demonstrated in a large-scale application, where it successfully recovers the initial conditions of the Navier-Stokes equations from solution data at positive times.
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