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Efficient real-time solvers for forward and inverse problems are essential in engineering and science applications. Machine learning surrogate models have emerged as promising alter- natives to traditional methods, offering substantially reduced computational time. Never- theless, these models typically demand extensive training datasets to achieve robust gen- eralization 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 model-constrained Tikhonov autoencoder neural network framework, called TAEN, capable of learning both forward and inverse surrogate models using a single arbitrary observational sample. We develop comprehensive theoretical foundations including forward and inverse inference error bounds for the proposed approach for linear cases. For compara- tive 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 with theoretical justification, which functions as a generative mechanism for exploring the train- ing data space, enabling effective training of both forward and inverse surrogate models even with 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 TAEN 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.
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Physics-Informed Deep Learning for Traffic State Estimation: A Survey and the Outlook
For its robust predictive power (compared to pure physics-based models) and sample-efficient training (compared to pure deep learning models), physics-informed deep learning (PIDL), a paradigm hybridizing physics-based models and deep neural networks (DNNs), has been booming in science and engineering fields. One key challenge of applying PIDL to various domains and problems lies in the design of a computational graph that integrates physics and DNNs. In other words, how the physics is encoded into DNNs and how the physics and data components are represented. In this paper, we offer an overview of a variety of architecture designs of PIDL computational graphs and how these structures are customized to traffic state estimation (TSE), a central problem in transportation engineering. When observation data, problem type, and goal vary, we demonstrate potential architectures of PIDL computational graphs and compare these variants using the same real-world dataset.
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- Award ID(s):
- 2038984
- PAR ID:
- 10447567
- Date Published:
- Journal Name:
- Algorithms
- Volume:
- 16
- Issue:
- 6
- ISSN:
- 1999-4893
- Page Range / eLocation ID:
- 305
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/a16060305
