[Paper Review] The efficacy and generalizability of conditional GANs for posterior inference in physics-based inverse problems

Deep Ray, Harisankar Ramaswamy+2 more|arXiv (Cornell University)|Feb 15, 2022

Generative Adversarial Networks and Image Synthesis4 citations

TL;DR

This paper proposes a conditional Wasserstein GAN (cWGAN) with a U-Net generator and conditional instance normalization (CIN) to enable efficient posterior sampling in physics-based Bayesian inverse problems governed by PDEs. By injecting latent variables at multiple scales, the method decouples latent dimension from measurement size, enables multi-scale stochasticity, and achieves accurate uncertainty quantification and generalization to out-of-distribution measurements across diverse PDE inverse problems including heat conduction and elasticity imaging.

ABSTRACT

In this work, we train conditional Wasserstein generative adversarial networks to effectively sample from the posterior of physics-based Bayesian inference problems. The generator is constructed using a U-Net architecture, with the latent information injected using conditional instance normalization. The former facilitates a multiscale inverse map, while the latter enables the decoupling of the latent space dimension from the dimension of the measurement, and introduces stochasticity at all scales of the U-Net. We solve PDE-based inverse problems to demonstrate the performance of our approach in quantifying the uncertainty in the inferred field. Further, we show the generator can learn inverse maps which are local in nature, which in turn promotes generalizability when testing with out-of-distribution samples.

Motivation & Objective

• Address the computational bottleneck of traditional Bayesian inference in PDE-based inverse problems by replacing slow MCMC sampling with deep generative modeling.
• Overcome limitations of deterministic regularization methods by enabling full posterior sampling and uncertainty quantification.
• Develop a scalable, generalizable deep learning framework for posterior inference that works across diverse physics-based inverse problems.
• Enable flexible latent space dimensioning independent of measurement or field dimensions through conditional instance normalization (CIN).
• Demonstrate robustness and generalization of the trained model on out-of-distribution (OOD) measurements, particularly in real experimental settings.

Proposed method

• Train a conditional Wasserstein GAN (cWGAN) using paired data from the joint distribution of inferred fields and measurements, generated via forward PDE solves with synthetic priors.
• Employ a residual block-based U-Net architecture as the generator to capture multiscale features from measurements and reconstruct the inferred field.
• Inject latent noise via conditional instance normalization (CIN) at multiple layers of the U-Net, enabling multi-scale stochasticity and decoupling of latent dimension from input/output dimensions.
• Use a simplified critic network instead of a conditional mini-batch critic, leveraging the theoretical convergence guarantees of WGANs for posterior statistics.
• Train the cWGAN end-to-end using a Wasserstein adversarial loss to ensure stable training and weak convergence of the generated posterior to the true posterior.
• Generate posterior samples by conditioning the trained generator on new measurements, bypassing MCMC and enabling fast, scalable inference.

Experimental results

Research questions

• RQ1 Can a cWGAN with CIN-based latent injection effectively learn the posterior distribution in diverse PDE-based inverse problems?
• RQ2 How does the use of CIN improve generalization and uncertainty quantification compared to standard cGANs with channel-wise latent stacking?
• RQ3 To what extent can a cWGAN trained on synthetic data generalize to real experimental measurements in elasticity imaging?
• RQ4 Does the learned inverse map exhibit local behavior that supports generalization to out-of-distribution (OOD) measurements?
• RQ5 Can the cWGAN provide reliable uncertainty estimates (e.g., standard deviation) that correlate with true reconstruction errors in physical fields?

Key findings

• The cWGAN with CIN successfully generates posterior samples across three distinct PDE inverse problems: heat conduction (initial condition and conductivity), and elasticity imaging.
• The model achieves a 7.7% error in estimating the vertical distance to the inclusion center and a 3% error in diameter estimation when tested on experimental ultrasound data, comparable to deterministic methods.
• The shear modulus of the inclusion was estimated at 13.3 kPa, with a 24.3% error compared to the measured 10.7 kPa, within the range of errors from deterministic approaches.
• Uncertainty (standard deviation) is highest along the boundary of inclusions, correctly reflecting regions of higher reconstruction ambiguity.
• The three most important samples (via RRQR) show minimal variability, indicating consistent posterior sampling and stable inference.
• The trained cWGAN generalizes well to OOD measurements, with theoretical and empirical evidence supporting the local nature of the learned inverse map as a key enabler of generalization.