# Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2403.03223 · HTML: https://mlsanity.com/paper/2403.03223/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 36 references that are in the map.
- First posted on arXiv 15 Feb 2024.

## Builds on (most cited first, 13 of 13)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations](https://mlsanity.com/paper/2107.10711/) (2021)
- [Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture](https://mlsanity.com/paper/2011.13511/) (2020)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Physics-informed neural networks for the shallow-water equations on the sphere](https://mlsanity.com/paper/2104.00615/) (2021)

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

- [Learnable Activation Functions in Physics-Informed Neural Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2411.15111/) (2024)
- [Adaptive Interface-PINNs (AdaI-PINNs): An Efficient Physics-informed Neural Networks Framework for Interface Problems](https://mlsanity.com/paper/2406.04626/) (2024)
- [iPINNER: An Iterative Physics-Informed Neural Network with Ensemble Kalman Filter](https://mlsanity.com/paper/2506.00731/) (2025)
- [Hard constraint learning approaches with trainable influence functions for evolutionary equations](https://mlsanity.com/paper/2502.17497/) (2025)
- [WellPINN: Accurate Well Representation for Transient Fluid Pressure Diffusion in Subsurface Reservoirs with Physics-Informed Neural Networks](https://mlsanity.com/paper/2507.09330/) (2025)
- [Enhancing PINN Accuracy for the RLW Equation: Adaptive and Conservative Approaches](https://mlsanity.com/paper/2511.11638/) (2025)
- [Residual Attention Physics-Informed Neural Networks for Robust Multiphysics Simulation of Steady-State Electrothermal Energy Systems](https://mlsanity.com/paper/2603.23578/) (2026)
- [Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data](https://mlsanity.com/paper/2605.20514/) (2026)
- [CI-PINN: Causal Integral Physics-Informed Neural Network for Solving Evolution Equations](https://mlsanity.com/paper/2609.36615/) (2026)

## Abstract (arXiv)

The use of deep learning methods in scientific computing represents a potential paradigm shift in engineering problem solving. One of the most prominent developments is Physics-Informed Neural Networks (PINNs), in which neural networks are trained to satisfy partial differential equations (PDEs). While this method shows promise, the standard version has been shown to struggle in accurately predicting the dynamic behavior of time-dependent problems. To address this challenge, methods have been proposed that decompose the time domain into multiple segments, employing a distinct neural network in each segment and directly incorporating continuity between them in the loss function of the minimization problem. In this work we introduce a method to exactly enforce continuity between successive time segments via a solution ansatz. This hard constrained sequential PINN (HCS-PINN) method is simple to implement and eliminates the need for any loss terms associated with temporal continuity. The method is tested for a number of benchmark problems involving both linear and non-linear PDEs. Examples include various first order time dependent problems in which traditional PINNs struggle, namely advection, Allen-Cahn, and Korteweg-de Vries equations. Furthermore, second and third order time-dependent problems are demonstrated via wave and Jerky dynamics examples, respectively. Notably, the Jerky dynamics problem is chaotic, making the problem especially sensitive to temporal accuracy. The numerical experiments conducted with the proposed method demonstrated superior convergence and accuracy over both traditional PINNs and the soft-constrained counterparts.