Solving inverse problems using normalizing flow prior: Application to optical spectra

We introduce a machine learning approach for solving ill-posed inverse problems, specifically addressing the Fredholm integral equation of the first kind. Harnessing the powerful capabilities of normalizing flows to approximate data distributions, combined with a robust probabilistic framework, our approach stands out by delivering robust solutions capable of handling high-level noises and out-of-distribution data and providing uncertainty estimation. A distinct feature lies in the unsupervised learning framework inherent in deep generative models, providing our approach with unparalleled flexibility across diverse experimental setups. This flexibility is exemplified through the successful application of our method to measured optical spectra.

Figure 1

Hundred random samples from (a) training data and (b) data distribution $p_X(\mathbf{x};\widehat{\psi })$. (c) Some solutions $\widehat{\mathbf{x}}$ of inverse problem with true $\mathbf{x}$'s using Algorithm lg1 and (d) corresponding reconstructions $A\widehat{\mathbf{x}}$'s and true $\mathbf{y}$'s. (All $1/{\tau }^{\text{op}}$ 's are scaled down by 300.)

Figure 2

(a) Solutions of the inverse problem using Algorithm lg1 from observations with different noise levels: (b) $\sigma =0.0001$, (c) $\sigma =0.001$, (d) $\sigma =0.01$, and (e) $\sigma =0.1$. (All $1/{\tau }^{\text{op}}$ 's are scaled down by 300.)

Figure 3

Sample means (blue solid) and error bars of two standard deviations (filled region) are shown with true $\mathbf{x}$'s (red dashed) for noise levels (a) $\sigma =0.01$ and (b) $\sigma =0.1$.

Figure 4

Left column: comparison of inference results of $I^2\chi$ obtained using NF (dashed lines) and MEM (dotted lines). Right column: experimental observations (solid curves) and reconstructions using results of NF (dashed lines), and MEM (dotted) of $1/{\tau }^{\text{op}}$. Each experiment is performed in different temperature setups: top ($T=100\mathrm{K}$), middle ($T=200\mathrm{K}$), and bottom ($T=300\mathrm{K}$). Samples (OD60, D82, and OPT96) are differentiated using the same color code in all figures.

Figure 5

Comparison of coupling constants obtained from inference results of $I^2\chi$ using NF and MEM are compared at different temperature setups: $T=$ 100 K, 200 K, and 300 K.