Good Things Come in Pairs: Paired Autoencoders for Inverse Problems

Inverse problems involve reconstructing unknown parameters from indirect measurements, prevalent in fields like geophysics, medical imaging, and remote sensing. Traditional methods such as Tikhonov regularization, Bayesian inference, and adjoint-based optimization have been effective but face limitations in scalability, robustness, and uncertainty quantification. Recent advances leverage deep neural networks, including end-to-end models, unrolled optimization, and generative approaches like VAEs and GANs, which improve efficiency but often lack interpretability and reliable uncertainty estimates. The need for methods that combine robustness, out-of-distribution detection, and probabilistic reasoning has driven the development of new frameworks like paired autoencoders.

Despite progress, existing deep learning solutions for inverse problems often suffer from poor generalization to unseen data, inability to quantify uncertainty, and sensitivity to noise. Classical methods are computationally expensive and ill-posed, requiring regularization and prior knowledge. The core challenge is to develop a scalable, likelihood-free approach that can produce high-fidelity reconstructions, detect out-of-distribution samples, and provide uncertainty estimates simultaneously. Addressing these issues is crucial for deploying inverse solutions in real-world scenarios where data variability and noise are unavoidable.

The paper introduces a paired autoencoder architecture that learns bidirectional mappings between data and model parameters in a shared latent space, enabling high-quality reconstructions without explicit likelihoods. It incorporates latent space refinement to enhance accuracy and introduces a variational extension for uncertainty quantification. This combination offers robustness against out-of-distribution data, improved interpretability, and probabilistic insights, representing a significant step beyond traditional deterministic or likelihood-based models. The framework is theoretically grounded, validated through extensive experiments, and addresses key limitations of prior methods.

• �� Encode observed data y and quantity of interest x into a shared latent space z using trained encoders;
• �� Decode z to reconstruct x and y, optimizing reconstruction loss and regularization terms;
• �� Use likelihood-free training to avoid explicit probability modeling, focusing on reconstruction fidelity;
• �� Refine latent variables via optimization to fit observed data more accurately;
• �� Extend with a variational autoencoder to learn the distribution over z, enabling sampling and uncertainty estimation;
• �� Validate through seismic imaging and image inpainting experiments, comparing with baseline methods, assessing robustness, accuracy, and out-of-distribution detection.

Experiments involved seismic data from synthetic models and real datasets, as well as natural images with added noise. Baselines included classical regularization, end-to-end neural networks, and unrolled optimization. Metrics such as reconstruction error, out-of-distribution detection accuracy, and uncertainty coverage were used. Hyperparameters like latent dimension and training epochs were tuned via ablation studies. The models were tested under varying noise levels and distribution shifts to evaluate robustness and generalization, demonstrating the superiority of the proposed framework.

In seismic imaging, the paired autoencoder improved spatial resolution by 15% over traditional methods, maintaining performance under high noise. In image inpainting, the reconstruction error decreased by 20%, with out-of-distribution detection accuracy reaching 92%. The variational extension provided credible uncertainty bounds with coverage exceeding 85%. Latent space refinement significantly enhanced inverse accuracy, especially in noisy scenarios, confirming robustness and practical utility. These results highlight the method’s ability to deliver reliable reconstructions and uncertainty estimates in challenging conditions.

This framework is applicable to geophysical exploration, medical diagnostics, and remote sensing, where data are noisy or incomplete. It enables high-fidelity reconstructions, anomaly detection, and uncertainty quantification, supporting decision-making in critical applications. Its likelihood-free nature simplifies training and deployment, making it suitable for large-scale, real-time inverse problems. Future integration with physical models and multi-modal data could further expand its industrial impact.

The approach depends on representative training data; performance may decline with unseen distributions or extreme noise. Latent space dimension selection requires careful tuning, impacting accuracy. Variational training increases computational load, limiting real-time use. Further research is needed to improve scalability, generalization, and reduce inference costs, especially for high-dimensional, complex systems.