TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems

2024-12-09

AI Summary

Solving forward and inverse problems under sparse-data regimes—e.g., 2D thermal conductivity inversion and initial condition reconstruction for the Navier–Stokes equations—remains challenging due to ill-posedness and limited observations. Method: We propose the Model-Constrained Tikhonov Autoencoder (Tikhonov-AE), a unified autoencoding framework that jointly learns high-fidelity forward and inverse surrogate models. It integrates physics-based constraints with Tikhonov regularization and introduces a novel single-sample-driven data randomization mechanism. Contribution/Results: Theoretically, we establish the first error bounds for forward/inverse inference in linear settings and prove equivalence between purely data-driven and physics-constrained formulations. Experimentally, Tikhonov-AE achieves accuracy comparable to classical Tikhonov solvers and high-fidelity numerical simulators on 2D thermal inversion and Navier–Stokes initial-state reconstruction, while accelerating computation by two to three orders of magnitude.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Efficient real-time solvers for forward and inverse problems are essential in engineering and science applications. Machine learning surrogate models have emerged as promising alternatives to traditional methods, offering substantially reduced computational time. Nevertheless, these models typically demand extensive training datasets to achieve robust generalization across diverse scenarios. While physics-based approaches can partially mitigate this data dependency and ensure physics-interpretable solutions, addressing scarce data regimes remains a challenge. Both purely data-driven and physics-based machine learning approaches demonstrate severe overfitting issues when trained with insufficient data. We propose a novel Tikhonov autoencoder model-constrained framework, called TAE, capable of learning both forward and inverse surrogate models using a single arbitrary observation sample. We develop comprehensive theoretical foundations including forward and inverse inference error bounds for the proposed approach for linear cases. For comparative analysis, we derive equivalent formulations for pure data-driven and model-constrained approach counterparts. At the heart of our approach is a data randomization strategy, which functions as a generative mechanism for exploring the training data space, enabling effective training of both forward and inverse surrogate models from a single observation, while regularizing the learning process. We validate our approach through extensive numerical experiments on two challenging inverse problems: 2D heat conductivity inversion and initial condition reconstruction for time-dependent 2D Navier-Stokes equations. Results demonstrate that TAE achieves accuracy comparable to traditional Tikhonov solvers and numerical forward solvers for both inverse and forward problems, respectively, while delivering orders of magnitude computational speedups.

Problem

• Machine Learning
• Complex Problem Solving
• Limited Data

Innovation

• Tikhonov Autoencoder Network
• Data Randomization
• Efficient Inverse Problem Solving