# DAE-PINN: A Physics-Informed Neural Network Model for Simulating Differential-Algebraic Equations with Application to Power Networks

arXiv: https://arxiv.org/abs/2109.04304 · HTML: https://mlsanity.com/paper/2109.04304/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 66 references that are in the map.
- First posted on arXiv 9 Sep 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Intelligent Time-Adaptive Transient Stability Assessment System](https://mlsanity.com/paper/1610.08235/) (2016)
- [Data-driven learning of non-autonomous systems](https://mlsanity.com/paper/2006.02392/) (2020)
- [A Bi-layered Parallel Training Architecture for Large-scale Convolutional Neural Networks](https://mlsanity.com/paper/1810.07742/) (2018)
- [A Kinetic Monte Carlo Approach for Simulating Cascading Transmission Line Failure](https://mlsanity.com/paper/1912.08081/) (2019)
- [Model reduction for a power grid model](https://mlsanity.com/paper/1912.12163/) (2019)

## Built on by (most cited first, 28 of 28)

- [Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems](https://mlsanity.com/paper/2406.19662/) (2024)
- [DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectories](https://mlsanity.com/paper/2202.07176/) (2022)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulators](https://mlsanity.com/paper/2301.12538/) (2023)
- [NSGA-PINN: A Multi-Objective Optimization Method for Physics-Informed Neural Network Training](https://mlsanity.com/paper/2303.02219/) (2023)
- [Physics-Informed Neural Networks for Time-Domain Simulations: Accuracy, Computational Cost, and Flexibility](https://mlsanity.com/paper/2303.08994/) (2023)
- [A multifidelity approach to continual learning for physical systems](https://mlsanity.com/paper/2304.03894/) (2023)
- [PINNSim: A Simulator for Power System Dynamics based on Physics-Informed Neural Networks](https://mlsanity.com/paper/2303.10256/) (2023)
- [A PINN Approach to Symbolic Differential Operator Discovery with Sparse Data](https://mlsanity.com/paper/2212.04630/) (2022)
- [Learning Neural Differential Algebraic Equations via Operator Splitting](https://mlsanity.com/paper/2403.12938/) (2024)
- [Physics-Constrained Machine Learning for Chemical Engineering](https://mlsanity.com/paper/2508.20649/) (2025)
- [A Simultaneous Approach for Training Neural Differential-Algebraic Systems of Equations](https://mlsanity.com/paper/2504.04665/) (2025)
- [Error estimation for physics-informed neural networks with implicit Runge-Kutta methods](https://mlsanity.com/paper/2401.05211/) (2024)
- [Data-driven identification of port-Hamiltonian DAE systems by Gaussian processes](https://mlsanity.com/paper/2406.18726/) (2024)
- [Neural Port-Hamiltonian Differential Algebraic Equations for Compositional Learning of Electrical Networks](https://mlsanity.com/paper/2412.11215/) (2024)
- [Physical Information Neural Networks for Solving High-index Differential-algebraic Equation Systems Based on Radau Methods](https://mlsanity.com/paper/2310.12846/) (2023)
- [B-LSTM-MIONet: Bayesian LSTM-based Neural Operators for Learning the Response of Complex Dynamical Systems to Length-Variant Multiple Input Functions](https://mlsanity.com/paper/2311.16519/) (2023)
- [Physics-informed Multiple-Input Operators for efficient dynamic response prediction of structures](https://mlsanity.com/paper/2505.07090/) (2025)
- [A comprehensive analysis of PINNs: Variants, Applications, and Challenges](https://mlsanity.com/paper/2505.22761/) (2025)
- [Physics-Guided Dimension Reduction for Simulation-Free Operator Learning of Stiff Differential-Algebraic Systems](https://mlsanity.com/paper/2604.19930/) (2026)
- [Quantum Neural Networks for Solving Power System Transient Simulation Problem](https://mlsanity.com/paper/2405.11427/) (2024)
- [DAE-KAN: A Kolmogorov-Arnold Network Model for High-Index Differential-Algebraic Equations](https://mlsanity.com/paper/2504.15806/) (2025)
- [Physics-Informed Neural Networks for Device and Circuit Modeling: A Case Study of NeuroSPICE](https://mlsanity.com/paper/2512.23624/) (2025)
- [Double Coupling Architecture and Training Method for Optimization Problems of Differential Algebraic Equations with Parameters](https://mlsanity.com/paper/2603.22724/) (2026)
- [Conformalized Quantum DeepONet Ensembles: Towards Scalable Operator Learning with Distribution-Free Guarantees](https://mlsanity.com/paper/2605.00330/) (2026)
- [Adaptive Distance-Aware Trunk Deep Operator Learning for Long-Span Roadway Bridges](https://mlsanity.com/paper/2606.20015/) (2026)
- [Event-Structured Physics-Informed Neural Networks for Differentiable Critical Clearing Boundaries](https://mlsanity.com/paper/2607.27681/) (2026)
- [A Unified Mamba--MoE Surrogate for Closed-Loop Simulation and Measurement-Window Forecasting of Inverter Transients](https://mlsanity.com/paper/2608.15051/) (2026)

## Abstract (arXiv)

Deep learning-based surrogate modeling is becoming a promising approach for learning and simulating dynamical systems. Deep-learning methods, however, find very challenging learning stiff dynamics. In this paper, we develop DAE-PINN, the first effective deep-learning framework for learning and simulating the solution trajectories of nonlinear differential-algebraic equations (DAE), which present a form of infinite stiffness and describe, for example, the dynamics of power networks. Our DAE-PINN bases its effectiveness on the synergy between implicit Runge-Kutta time-stepping schemes (designed specifically for solving DAEs) and physics-informed neural networks (PINN) (deep neural networks that we train to satisfy the dynamics of the underlying problem). Furthermore, our framework (i) enforces the neural network to satisfy the DAEs as (approximate) hard constraints using a penalty-based method and (ii) enables simulating DAEs for long-time horizons. We showcase the effectiveness and accuracy of DAE-PINN by learning and simulating the solution trajectories of a three-bus power network.