Analyzing Inverse Problems with Invertible Neural Networks (1808.04730v3)

Abstract: In many tasks, in particular in natural science, the goal is to determine hidden system parameters from a set of measurements. Often, the forward process from parameter- to measurement-space is a well-defined function, whereas the inverse problem is ambiguous: one measurement may map to multiple different sets of parameters. In this setting, the posterior parameter distribution, conditioned on an input measurement, has to be determined. We argue that a particular class of neural networks is well suited for this task -- so-called Invertible Neural Networks (INNs). Although INNs are not new, they have, so far, received little attention in literature. While classical neural networks attempt to solve the ambiguous inverse problem directly, INNs are able to learn it jointly with the well-defined forward process, using additional latent output variables to capture the information otherwise lost. Given a specific measurement and sampled latent variables, the inverse pass of the INN provides a full distribution over parameter space. We verify experimentally, on artificial data and real-world problems from astrophysics and medicine, that INNs are a powerful analysis tool to find multi-modalities in parameter space, to uncover parameter correlations, and to identify unrecoverable parameters.

• The paper introduces a novel INN-based framework that accurately estimates the full posterior distribution of hidden parameters in inverse problems.
• It employs bijective mappings with tractable Jacobians to capture multimodal parameter correlations and quantify uncertainties.
• Experimental results on synthetic and real-world datasets validate its efficiency and superiority over methods like ABC and conditional VAEs.

Analyzing Inverse Problems with Invertible Neural Networks

This paper presents an in-depth examination of inverse problems through the lens of Invertible Neural Networks (INNs). It highlights the utility of INNs in determining hidden system parameters from observable data—a common challenge in scientific domains like astrophysics and medicine.

Problem Statement and Methodology

In many scientific applications, the forward process—mapping hidden parameters to observable outcomes—is well understood, but the inverse process remains ambiguous due to information loss and degeneracy. The primary aim of this paper is to fully characterize the posterior distribution of hidden parameters conditioned on observed data, thus quantifying inherent uncertainties and revealing parameter correlations or irretrievability.

Invertible neural networks are proposed as a solution because of their properties: they offer bijective mappings, computationally feasible forward and inverse operations, and tractable Jacobians for calculating posterior probabilities. Unlike traditional networks that directly address inverse problems, INNs implicitly learn the inverse process by focusing on the simpler and more defined forward process. They incorporate latent variables to capture the missing information, thus enabling the reconstruction of the posterior over the parameter space given new measurements.

Key Contributions

1. Posterior Estimation: The research demonstrates, both theoretically and through experimental validation, that INNs can accurately estimate the full posterior of an inverse problem, capturing multimodalities and parameter correlations.
2. Efficiency and Representational Power: The constraints imposed by the invertible architecture do not undermine the network's ability to represent complex data transformations.
3. Enhanced Training: By combining forward training with unsupervised backward training, INNs achieve improved performance on datasets with finite observations.
4. Comparison with Other Methods: The paper compares INNs favorably against methods such as Approximate Bayesian Computation (ABC) and conditional VAEs, emphasizing their robustness in identifying parameter relations and uncertainties.

Experimental Validation

The efficacy of the proposed approach is validated across synthetic and real-world datasets. In synthetic mixtures, INNs effectively modeled the posterior without mode collapse. For inverse kinematics, they performed comparably to ground truth samples obtained from ABC. In medical and astrophysical applications, INNs identified parameter dependencies and unrecoverable dimensions, providing richer insights than other comparative methods.

Implications and Future Developments

The implications of this work are significant for scientific fields requiring precise parameter estimations from indirect measurements. By accurately modeling the full posterior distribution, INNs contribute to improved understanding and decision-making processes. The ability to capture multimodality and parameter correlations has practical implications in designing better experiments and interpreting complex systems.

Looking ahead, there is potential to refine INNs further by exploring various invertible architectures and integrating cycle consistency losses. Additionally, scaling these methods to handle higher dimensional datasets remains a promising area for future research, potentially enhancing the efficacy of INNs in even more complex inverse problem scenarios.

This paper thoroughly elucidates the capabilities of Invertible Neural Networks in solving inverse problems, offering a compelling alternative to traditional approaches in data-driven scientific investigation.