# Physics-constrained Deep Learning for Robust Inverse ECG Modeling

arXiv: https://arxiv.org/abs/2107.12780 · HTML: https://mlsanity.com/paper/2107.12780/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 87 references that are in the map.
- First posted on arXiv 26 Jul 2021.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Improving Generalization Performance by Switching from Adam to SGD](https://mlsanity.com/paper/1712.07628/) (2017)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Physics-Guided Machine Learning for Scientific Discovery: An Application in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/2001.11086/) (2020)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [Physics-Guided Architecture (PGA) of Neural Networks for Quantifying Uncertainty in Lake Temperature Modeling](https://mlsanity.com/paper/1911.02682/) (2019)
- [Benchmarking State-of-the-Art Deep Learning Software Tools](https://mlsanity.com/paper/1608.07249/) (2016)
- [DropSample: A New Training Method to Enhance Deep Convolutional Neural Networks for Large-Scale Unconstrained Handwritten Chinese Character Recognition](https://mlsanity.com/paper/1505.05354/) (2015)

## Built on by (most cited first, 12 of 12)

- [Towards Enabling Cardiac Digital Twins of Myocardial Infarction Using Deep Computational Models for Inverse Inference](https://mlsanity.com/paper/2307.04421/) (2023)
- [Hierarchical Deep Learning with Generative Adversarial Network for Automatic Cardiac Diagnosis from ECG Signals](https://mlsanity.com/paper/2210.11408/) (2022)
- [Solving the Inverse Problem of Electrocardiography for Cardiac Digital Twins: A Survey](https://mlsanity.com/paper/2406.11445/) (2024)
- [MUSE-Net: Missingness-aware mUlti-branching Self-attention Encoder for Irregular Longitudinal Electronic Health Records](https://mlsanity.com/paper/2407.00840/) (2024)
- [Geometry-aware Active Learning of Spatiotemporal Dynamic Systems](https://mlsanity.com/paper/2504.19012/) (2025)
- [Full-field surrogate modeling of cardiac function encoding geometric variability](https://mlsanity.com/paper/2504.20479/) (2025)
- [The Effect of Different Optimization Strategies to Physics-Constrained Deep Learning for Soil Moisture Estimation](https://mlsanity.com/paper/2403.08154/) (2024)
- [Physics-augmented Multi-task Gaussian Process for Modeling Spatiotemporal Dynamics](https://mlsanity.com/paper/2510.13601/) (2025)
- [Physically Informed Synchronic-adaptive Learning for Industrial Systems Modeling in Heterogeneous Media with Unavailable Time-varying Interface](https://mlsanity.com/paper/2401.14609/) (2024)
- [Chain of Flow: ECG-Conditioned 4D Cardiac Cine Generation from Patient-Specific Anatomical Anchor](https://mlsanity.com/paper/2602.22919/) (2026)
- [Multi-Modal Machine Learning for Breast Cancer Recurrence Prediction](https://mlsanity.com/paper/2606.02892/) (2026)
- [P-K-GCN: Physics-augmented Koopman-enhanced Graph Convolutional Network for Deep Spatiotemporal Super-resolution](https://mlsanity.com/paper/2606.19303/) (2026)

## Abstract (arXiv)

The rapid developments in advanced sensing and imaging bring about a data-rich environment, facilitating the effective modeling, monitoring, and control of complex systems. For example, the body-sensor network captures multi-channel information pertinent to the electrical activity of the heart (i.e., electrocardiograms (ECG)), which enables medical scientists to monitor and detect abnormal cardiac conditions. However, the high-dimensional sensing data are generally complexly structured and realizing the full data potential depends to a great extent on advanced analytical and predictive methods. This paper presents a physics-constrained deep learning (P-DL) framework for high-dimensional inverse ECG modeling. This method integrates the physical laws of the complex system with the advanced deep learning infrastructure for effective prediction of the system dynamics. The proposed P-DL approach is implemented to solve the inverse ECG model and predict the time-varying distribution of electric potentials in the heart from the ECG data measured by the body-surface sensor network. Experimental results show that the proposed P-DL method significantly outperforms existing methods that are commonly used in current practice.