# Interpreting and generalizing deep learning in physics-based problems with functional linear models

arXiv: https://arxiv.org/abs/2307.04569 · HTML: https://mlsanity.com/paper/2307.04569/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 100 references that are in the map.
- First posted on arXiv 10 Jul 2023.

## Builds on (most cited first, 30 of 33)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Generalized Out-of-Distribution Detection: A Survey](https://mlsanity.com/paper/2110.11334/) (2021)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [Learning Confidence for Out-of-Distribution Detection in Neural Networks](https://mlsanity.com/paper/1802.04865/) (2018)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Transparency of Deep Neural Networks for Medical Image Analysis: A Review of Interpretability Methods](https://mlsanity.com/paper/2111.02398/) (2021)
- [LassoNet: A Neural Network with Feature Sparsity](https://mlsanity.com/paper/1907.12207/) (2019)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Unsupervised discovery of interpretable hyperelastic constitutive laws](https://mlsanity.com/paper/2010.13496/) (2020)
- [Convolutional-network models to predict wall-bounded turbulence from wall quantities](https://mlsanity.com/paper/2006.12483/) (2020)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural network](https://mlsanity.com/paper/2108.11985/) (2021)
- [Super-Resolution Analysis via Machine Learning: A Survey for Fluid Flows](https://mlsanity.com/paper/2301.10937/) (2023)
- [Sparsity-promoting dynamic mode decomposition](https://mlsanity.com/paper/1309.4165/) (2013)
- [On transfer learning of neural networks using bi-fidelity data for uncertainty propagation](https://mlsanity.com/paper/2002.04495/) (2020)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [Sparse Neural Additive Model: Interpretable Deep Learning with Feature Selection via Group Sparsity](https://mlsanity.com/paper/2202.12482/) (2022)
- [Designing Inherently Interpretable Machine Learning Models](https://mlsanity.com/paper/2111.01743/) (2021)
- [Quantifying Model Complexity via Functional Decomposition for Better Post-Hoc Interpretability](https://mlsanity.com/paper/1904.03867/) (2019)
- [Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)](https://mlsanity.com/paper/2106.01510/) (2021)
- [A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurements](https://mlsanity.com/paper/2204.00205/) (2022)
- [Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils](https://mlsanity.com/paper/2302.00807/) (2023)
- [Neural Network Training Using ℓ1-Regularization and Bi-fidelity Data](https://mlsanity.com/paper/2105.13011/) (2021)
- [Towards out of distribution generalization for problems in mechanics](https://mlsanity.com/paper/2206.14917/) (2022)
- [Interpretable models for extrapolation in scientific machine learning](https://mlsanity.com/paper/2212.10283/) (2022)
- [Forecasting subcritical cylinder wakes with Fourier Neural Operators](https://mlsanity.com/paper/2301.08290/) (2023)

## Built on by (most cited first, 9 of 9)

- [ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification](https://mlsanity.com/paper/2410.16528/) (2024)
- [Additive-feature-attribution methods: a review on explainable artificial intelligence for fluid dynamics and heat transfer](https://mlsanity.com/paper/2409.11992/) (2024)
- [Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learning](https://mlsanity.com/paper/2603.23854/) (2026)
- [SLE-FNO: Single-Layer Extensions for Task-Agnostic Continual Learning in Fourier Neural Operators](https://mlsanity.com/paper/2603.20410/) (2026)
- [Segmenting mechanically heterogeneous domains via unsupervised learning](https://mlsanity.com/paper/2308.15697/) (2023)
- [Interpretable Diagnostics and Adaptive Data Assimilation for Neural ODEs via Discrete Empirical Interpolation](https://mlsanity.com/paper/2510.21852/) (2025)
- [DreamSAC: Learning Hamiltonian World Models via Symmetry Exploration](https://mlsanity.com/paper/2603.07545/) (2026)
- [Machine Learning-Guided Design of Non-Reciprocal and Asymmetric Elastic Chiral Metamaterials](https://mlsanity.com/paper/2404.13215/) (2024)
- [Evidence-Guided Neural Architecture Selection under Uncertainty for Subject-Specific Blood Glucose Forecasting](https://mlsanity.com/paper/2606.05373/) (2026)

## Abstract (arXiv)

Although deep learning has achieved remarkable success in various scientific machine learning applications, its opaque nature poses concerns regarding interpretability and generalization capabilities beyond the training data. Interpretability is crucial and often desired in modeling physical systems. Moreover, acquiring extensive datasets that encompass the entire range of input features is challenging in many physics-based learning tasks, leading to increased errors when encountering out-of-distribution (OOD) data. In this work, motivated by the field of functional data analysis (FDA), we propose generalized functional linear models as an interpretable surrogate for a trained deep learning model. We demonstrate that our model could be trained either based on a trained neural network (post-hoc interpretation) or directly from training data (interpretable operator learning). A library of generalized functional linear models with different kernel functions is considered and sparse regression is used to discover an interpretable surrogate model that could be analytically presented. We present test cases in solid mechanics, fluid mechanics, and transport. Our results demonstrate that our model can achieve comparable accuracy to deep learning and can improve OOD generalization while providing more transparency and interpretability. Our study underscores the significance of interpretable representation in scientific machine learning and showcases the potential of functional linear models as a tool for interpreting and generalizing deep learning.