# Development and Validation of a Physics-Guided Machine Learning Extrapolation Framework Using a Classical Transient Diffusion Benchmark

arXiv: https://arxiv.org/abs/2609.09912 · HTML: https://mlsanity.com/paper/2609.09912/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 35 references that are in the map.
- First posted on arXiv 9 Sep 2026.

## Builds on (most cited first, 12 of 12)

- [Speech Recognition with Deep Recurrent Neural Networks](https://mlsanity.com/paper/1303.5778/) (2013)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [How Neural Networks Extrapolate: From Feedforward to Graph Neural Networks](https://mlsanity.com/paper/2009.11848/) (2020)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [A Probabilistic Graphical Model Foundation for Enabling Predictive Digital Twins at Scale](https://mlsanity.com/paper/2012.05841/) (2020)
- [A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications](https://mlsanity.com/paper/2010.02011/) (2020)
- [Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structures](https://mlsanity.com/paper/2002.10253/) (2020)
- [DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation](https://mlsanity.com/paper/2012.02681/) (2020)
- [Deep encoder-decoder hierarchical convolutional neural networks for conjugate heat transfer surrogate modeling](https://mlsanity.com/paper/2311.17068/) (2023)
- [Deep Convolutional Architectures for Extrapolative Forecast in Time-dependent Flow Problems](https://mlsanity.com/paper/2209.09651/) (2022)

## Abstract (arXiv)

Machine learning models used in engineering are typically trained within limited operating ranges, yet reliable predictions are often required beyond these domains. Consequently, the primary challenge is extrapolation rather than interpolation. Rigorous validation is hindered by the scarcity of data outside the training range. To address this limitation, a novel extrapolation framework is integrated with established machine learning architectures to enable accurate and physically consistent predictions beyond the training domain. The framework is established by systematically evaluating two physics-guided architectures: a Bidirectional Long Short-Term Memory (BiLSTM) network and a Physics-Informed Neural Network (PINN). A classical one-dimensional transient diffusion problem is adopted as a benchmark because its exact analytical solution provides unlimited, reliable data across the spatio-temporal domain, enabling rigorous quantitative validation. The problem is particularly challenging because the solution evolves from an initial singularity through a strongly nonlinear transient regime before approaching a steady-state linear profile. When training data are confined to an intermediate portion of this evolution, backward extrapolation toward the singularity becomes especially demanding. To improve reliability, physics-guided coordinate transformations, boundary-aware learning strategies, and stability-enhancing temporal marching are incorporated. Extrapolation is evaluated using a train-predict-validate-extend strategy, in which validated predictions are recursively added to the training set to progressively extend the prediction horizon. The results demonstrate accurate and physically consistent predictions beyond the training domain, highlighting the framework's potential for engineering applications where data availability is limited.