# TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems Source: [TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems]() Collection index: [llms.txt]() Data updated: 2026-05-05T05:00:04.460Z Authors: Hai V. Nguyen, Tan Bui-Thanh, Clint Dawson Published: 2024-12-09T21:36:42.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Forward Problem (PtO) on 2D Navier-Stokes (test)](): Forward Problem (PtO); 2D Navier-Stokes (test); Average Relative Error: 1.45. - [Forward mapping on 2D Heat Equation 16 x 16 finite element mesh (test)](): Forward mapping; 2D Heat Equation 16 x 16 finite element mesh (test); Average Relative Error: 1.22. - [Inverse Mapping on 2D Heat Equation 16 x 16 finite element mesh (test)](): Inverse mapping; 2D Heat Equation 16 x 16 finite element mesh (test); Average Relative Error: 45.03. - [Inverse Problem on 2D Navier-Stokes (test)](): Inverse Problem; 2D Navier-Stokes (test); Average Relative Error: 24.54. ## Original sources and implementation [Original paper]() [PDF]()