# Deep neural network enabled corrective source term approach to hybrid analysis and modeling

arXiv: https://arxiv.org/abs/2105.11521 · HTML: https://mlsanity.com/paper/2105.11521/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 62 references that are in the map.
- First posted on arXiv 24 May 2021.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Inception-v4, Inception-ResNet and the Impact of Residual Connections on Learning](https://mlsanity.com/paper/1602.07261/) (2016)
- [Threat of Adversarial Attacks on Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/1801.00553/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Adversarial Examples: Attacks and Defenses for Deep Learning](https://mlsanity.com/paper/1712.07107/) (2017)
- [Adversarial Attacks and Defenses in Images, Graphs and Text: A Review](https://mlsanity.com/paper/1909.08072/) (2019)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Digital Twin: Values, Challenges and Enablers](https://mlsanity.com/paper/1910.01719/) (2019)
- [Speaker Recognition Based on Deep Learning: An Overview](https://mlsanity.com/paper/2012.00931/) (2020)
- [A Probabilistic Graphical Model Foundation for Enabling Predictive Digital Twins at Scale](https://mlsanity.com/paper/2012.05841/) (2020)
- [tempoGAN: A Temporally Coherent, Volumetric GAN for Super-resolution Fluid Flow](https://mlsanity.com/paper/1801.09710/) (2018)
- [Physics guided machine learning using simplified theories](https://mlsanity.com/paper/2012.13343/) (2020)
- [The role of surrogate models in the development of digital twins of dynamic systems](https://mlsanity.com/paper/2001.09292/) (2020)
- [Data-driven modelling of nonlinear spatio-temporal fluid flows using a deep convolutional generative adversarial network](https://mlsanity.com/paper/2004.00707/) (2020)
- [Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution](https://mlsanity.com/paper/2103.14629/) (2021)
- [An advanced hybrid deep adversarial autoencoder for parameterized nonlinear fluid flow modelling](https://mlsanity.com/paper/2003.10547/) (2020)
- [Joint Gaussian Processes for Biophysical Parameter Retrieval](https://mlsanity.com/paper/1711.05197/) (2017)
- [A non-intrusive reduced order modeling framework for quasi-geostrophic turbulence](https://mlsanity.com/paper/1906.11617/) (2019)

## Built on by (most cited first, 16 of 16)

- [Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONets](https://mlsanity.com/paper/2204.00997/) (2022)
- [Physics guided neural networks for modelling of non-linear dynamics](https://mlsanity.com/paper/2205.06858/) (2022)
- [Multiscale Neural Operator: Learning Fast and Grid-independent PDE Solvers](https://mlsanity.com/paper/2207.11417/) (2022)
- [A Thermodynamics-informed Active Learning Approach to Perception and Reasoning about Fluids](https://mlsanity.com/paper/2203.05775/) (2022)
- [A singular Riemannian geometry approach to Deep Neural Networks II. Reconstruction of 1-D equivalence classes](https://mlsanity.com/paper/2112.10583/) (2021)
- [Digital Twins in Wind Energy: Emerging Technologies and Industry-Informed Future Directions](https://mlsanity.com/paper/2304.11405/) (2023)
- [Combining physics-based and data-driven techniques for reliable hybrid analysis and modeling using the corrective source term approach](https://mlsanity.com/paper/2206.03451/) (2022)
- [A novel corrective-source term approach to modeling unknown physics in aluminum extraction process](https://mlsanity.com/paper/2209.10861/) (2022)
- [Artificial intelligence-driven digital twin of a modern house demonstrated in virtual reality](https://mlsanity.com/paper/2212.07102/) (2022)
- [Sparse deep neural networks for modeling aluminum electrolysis dynamics](https://mlsanity.com/paper/2209.05832/) (2022)
- [INC: An Indirect Neural Corrector for Auto-Regressive Hybrid PDE Solvers](https://mlsanity.com/paper/2511.12764/) (2025)
- [Hybrid Modeling, Sim-to-Real Reinforcement Learning, and Large Language Model Driven Control for Digital Twins](https://mlsanity.com/paper/2510.23882/) (2025)
- [Supplementation of deep neural networks with simplified physics-based features to increase model prediction accuracy](https://mlsanity.com/paper/2204.06764/) (2022)
- [A Bi-fidelity DeepONet Approach for Modeling Uncertain and Degrading Hysteretic Systems](https://mlsanity.com/paper/2304.12609/) (2023)
- [Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Data](https://mlsanity.com/paper/2505.11308/) (2025)
- [RECAST: A Machine-Learning Framework for Correction and Super-Resolution of Coarse-Grid PDE Solvers](https://mlsanity.com/paper/2608.11572/) (2026)

## Abstract (arXiv)

In this work, we introduce, justify and demonstrate the Corrective Source Term Approach (CoSTA) -- a novel approach to Hybrid Analysis and Modeling (HAM). The objective of HAM is to combine physics-based modeling (PBM) and data-driven modeling (DDM) to create generalizable, trustworthy, accurate, computationally efficient and self-evolving models. CoSTA achieves this objective by augmenting the governing equation of a PBM model with a corrective source term generated using a deep neural network. In a series of numerical experiments on one-dimensional heat diffusion, CoSTA is found to outperform comparable DDM and PBM models in terms of accuracy -- often reducing predictive errors by several orders of magnitude -- while also generalizing better than pure DDM. Due to its flexible but solid theoretical foundation, CoSTA provides a modular framework for leveraging novel developments within both PBM and DDM. Its theoretical foundation also ensures that CoSTA can be used to model any system governed by (deterministic) partial differential equations. Moreover, CoSTA facilitates interpretation of the DNN-generated source term within the context of PBM, which results in improved explainability of the DNN. These factors make CoSTA a potential door-opener for data-driven techniques to enter high-stakes applications previously reserved for pure PBM.