NestyNet. II. 来自科学神经代理的相干函数空间后验（或如何避免昂贵的MCMC）
NestyNet. II. Coherent Function-Space Posteriors from Scientific Neural Surrogates (or How to Avoid Expensive MCMC)
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中文总结 AI 辅助
本文针对经二阶优化训练的科学神经代理，提出一种从其构建相干函数空间后验的高效方法，可避免昂贵的MCMC，通过低维不确定度坐标捕获后验方差，实现快速的不确定度传播。
中文摘要 AI 辅助
科学分析越来越多地使用灵活的神经网络，但它们的数千个相关参数使得解释相关不确定性颇具挑战。本文针对经二阶优化训练的科学神经代理，为拟合函数本身构建低维后验。通过对随机残差行线性化拟合过程，得到测量到函数的变换（线性映射），该线性映射由收敛的雅可比矩阵和高斯-牛顿曲率组装而成，可将测量扰动转换为重新拟合会产生的函数扰动。其主导奇异函数定义了相干形变模式，独立的高斯系数随后生成平滑函数采样，使得包括需要采样导数或积分的任何导出量都能继承后验。该构建区分了重复实验协方差与局部高斯-牛顿/拉普拉斯后验，并将两者传播到科学关注的相关量。结果依赖于拟合的声明选择（架构、超参数、活动集和优化分支），且每次拟合都通过检查进一步优化步骤是否会改变拟合预测，且该改变小于测量误差的选定小比例来确保收敛。本文的主要示例是针对模拟恒星盘的800参数相空间分布函数拟合，四个不确定度坐标（比拟合参数少两个数量级，且在力基细化下稳定）捕获了99%的垂直力后验方差，4000次相干采样在0.8秒内完成了力、总密度、表面密度和频率计算的传播。该方法为依赖导数的科学推断提供了一种高效的不确定度传播途径。
英文摘要
Scientific analyses increasingly use flexible neural networks, but their thousands of correlated parameters make it challenging to interpret the associated uncertainties. Here we develop a low-dimensional posterior for the fitted function itself, for scientific neural surrogates trained with second-order optimization. Linearizing the fitting procedure with respect to the randomized residual rows gives a measurement-to-function transport, the linear map, assembled from the converged Jacobians and Gauss--Newton curvature, that carries measurement perturbations into the function perturbations that refitting would produce. Its leading singular functions define coherent deformation modes. Independent Gaussian coefficients then generate smooth function draws, so that any derived quantity, including those requiring derivatives or integrals of the draw, inherits the posterior. The construction distinguishes repeated-experiment covariance from the local Gauss--Newton/Laplace posterior and propagates both to correlated quantities of scientific interest. The result is conditional on the fit's declared choices (architecture, hyperparameters, active set, and optimization branch), and every fit is certified as converged by checking that a further optimization step would change the fitted predictions by less than a chosen small fraction of the measurement errors. Our primary example is an 800-parameter phase-space distribution function fit for a mock stellar disk. Four uncertainty coordinates, two orders of magnitude fewer than the fitted parameters and stable under refinement of the force basis, capture 99\% of the vertical-force posterior variance, and 4000 coherent draws propagate through the force, total-density, surface-density, and frequency calculations in 0.8s. The method provides a highly efficient route to uncertainty propagation for derivative-dependent scientific inference.
