# A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurements

arXiv: https://arxiv.org/abs/2204.00205 · HTML: https://mlsanity.com/paper/2204.00205/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 62 references that are in the map.
- First posted on arXiv 1 Apr 2022.

## Builds on (most cited first, 22 of 22)

- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)
- [EikoNet: Solving the Eikonal equation with Deep Neural Networks](https://mlsanity.com/paper/2004.00361/) (2020)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural network](https://mlsanity.com/paper/2108.11985/) (2021)
- [A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials](https://mlsanity.com/paper/1807.09829/) (2018)
- [Unsupervised Deep Learning Algorithm for PDE-based Forward and Inverse Problems](https://mlsanity.com/paper/1904.05417/) (2019)
- [Learning Soft Tissue Behavior of Organs for Surgical Navigation with Convolutional Neural Networks](https://mlsanity.com/paper/1904.00722/) (2019)
- [Data-driven Modeling of the Mechanical Behavior of Anisotropic Soft Biological Tissue](https://mlsanity.com/paper/2107.05388/) (2021)

## Built on by (most cited first, 20 of 20)

- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [MgNO: Efficient Parameterization of Linear Operators via Multigrid](https://mlsanity.com/paper/2310.19809/) (2023)
- [Domain Agnostic Fourier Neural Operators](https://mlsanity.com/paper/2305.00478/) (2023)
- [INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservation](https://mlsanity.com/paper/2212.14365/) (2022)
- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Recent Advances and Applications of Machine Learning in Experimental Solid Mechanics: A Review](https://mlsanity.com/paper/2303.07647/) (2023)
- [Deep Neural Operator Enabled Digital Twin Modeling for Additive Manufacturing](https://mlsanity.com/paper/2405.09572/) (2024)
- [Interpreting and generalizing deep learning in physics-based problems with functional linear models](https://mlsanity.com/paper/2307.04569/) (2023)
- [Generative Hyperelasticity with Physics-Informed Probabilistic Diffusion Fields](https://mlsanity.com/paper/2310.03745/) (2023)
- [Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material models](https://mlsanity.com/paper/2602.17750/) (2026)
- [Deep learning for full-field ultrasonic characterization](https://mlsanity.com/paper/2301.02378/) (2023)
- [Heterogeneous Peridynamic Neural Operators: Discover Biotissue Constitutive Law and Microstructure From Digital Image Correlation Measurements](https://mlsanity.com/paper/2403.18597/) (2024)
- [Physics-Informed Machine Learning in Biomedical Science and Engineering](https://mlsanity.com/paper/2510.05433/) (2025)
- [Towards a unified nonlocal, peridynamics framework for the coarse-graining of molecular dynamics data with fractures](https://mlsanity.com/paper/2301.04540/) (2023)
- [Self-composing neural operators for high-frequency and multiscale PDE surrogates](https://mlsanity.com/paper/2508.20650/) (2025)
- [Segmenting mechanically heterogeneous domains via unsupervised learning](https://mlsanity.com/paper/2308.15697/) (2023)
- [Explainable Hierarchical Deep Learning Neural Networks (Ex-HiDeNN)](https://mlsanity.com/paper/2507.05498/) (2025)
- [Uncertainty quantification in mechanics: A unified Bayesian perspective](https://mlsanity.com/paper/2607.18734/) (2026)
- [pyALDIC: A Python Implementation of Augmented Lagrangian Digital Image Correlation with a GUI, Adaptive Meshing, and Mask-Aware Subset Splitting](https://mlsanity.com/paper/2607.22755/) (2026)

## Abstract (arXiv)

We present a data-driven workflow to biological tissue modeling, which aims to predict the displacement field based on digital image correlation (DIC) measurements under unseen loading scenarios, without postulating a specific constitutive model form nor possessing knowledges on the material microstructure. To this end, a material database is constructed from the DIC displacement tracking measurements of multiple biaxial stretching protocols on a porcine tricuspid valve anterior leaflet, with which we build a neural operator learning model. The material response is modeled as a solution operator from the loading to the resultant displacement field, with the material microstructure properties learned implicitly from the data and naturally embedded in the network parameters. Using various combinations of loading protocols, we compare the predictivity of this framework with finite element analysis based on the phenomenological Fung-type model. From in-distribution tests, the predictivity of our approach presents good generalizability to different loading conditions and outperforms the conventional constitutive modeling at approximately one order of magnitude. When tested on out-of-distribution loading ratios, the neural operator learning approach becomes less effective. To improve the generalizability of our framework, we propose a physics-guided neural operator learning model via imposing partial physics knowledge. This method is shown to improve the model's extrapolative performance in the small-deformation regime. Our results demonstrate that with sufficient data coverage and/or guidance from partial physics constraints, the data-driven approach can be a more effective method for modeling biological materials than the traditional constitutive modeling.