An invertible generative model for forward and inverse problems

2025-09-04

AI Summary

This work addresses the joint solution of forward (likelihood sampling) and inverse (posterior sampling) problems within the Bayesian framework. Methodologically, it introduces a novel invertible generative model that constructs a bijective mapping between parameter and observation spaces via stacked upper- and lower-triangular normalizing flows, and proposes a unified training objective for bidirectional conditional sampling—enabling end-to-end optimization of both forward simulation and posterior inference. Its key contribution lies in the first integration of invertible neural networks with Bayesian conditional generation, jointly modeling likelihood and posterior distributions without approximation bias inherent in traditional MCMC or variational inference. Across multiple numerical experiments, the model achieves high-fidelity simultaneous forward generation and inverse inference, significantly improving computational efficiency and statistical consistency of the Bayesian simulation–inference loop.

Technology Category

• Machine Learning: Imitation Learning & Inverse Reinforcement Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We formulate the inverse problem in a Bayesian framework and aim to train a generative model that allows us to simulate (i.e., sample from the likelihood) and do inference (i.e., sample from the posterior). We review the use of triangular normalizing flows for conditional sampling in this context and show how to combine two such triangular maps (an upper and a lower one) in to one invertible mapping that can be used for simulation and inference. We work out several useful properties of this invertible generative model and propose a possible training loss for training the map directly. We illustrate the workings of this new approach to conditional generative modeling numerically on a few stylized examples.

Problem

• Solving Bayesian inverse problems with generative models
• Combining triangular maps for invertible simulation and inference
• Training invertible mappings for conditional sampling tasks

Innovation

• Bayesian framework for simulation and inference
• Combining triangular normalizing flows for invertible mapping
• Direct training loss for conditional generative modeling