Skip to main content
OSTI.GOVU.S. Department of Energy Office of Scientific and Technical Information
U.S. Department of Energy
Office of Scientific and Technical Information
 

Exact enforcement of temporal continuity in sequential physics-informed neural networks

Journal Article · · Computer Methods in Applied Mechanics and Engineering
 [1];  [1]
  1. Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
+ Show Author Affiliations

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. Finally, the numerical experiments conducted with the proposed method demonstrated superior convergence and accuracy over both traditional PINNs and the soft-constrained counterparts.

Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
AC52-07NA27344
OSTI ID:
2429369
Report Number(s):
LLNL--JRNL-859490; 1090524
Journal Information:
Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 430; ISSN 0045-7825
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English

References (29)

Deep learning phase‐field model for brittle fractures
  • Ghaffari Motlagh, Yousef; Jimack, Peter K.; de Borst, René
  • International Journal for Numerical Methods in Engineering, Vol. 124, Issue 3 https://doi.org/10.1002/nme.7135
journal October 2022
On the limited memory BFGS method for large scale optimization journal August 1989
Physics-informed neural networks (PINNs) for fluid mechanics: a review journal December 2021
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next journal July 2022
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening journal June 1979
Physics informed neural networks reveal valid models for reactive diffusion of volatiles through paper journal March 2024
Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks journal February 2022
A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations journal February 2022
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations journal February 2019
B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data journal January 2021
A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks journal June 2021
When and why PINNs fail to train: A neural tangent kernel perspective journal January 2022
Physics-informed neural networks for the shallow-water equations on the sphere journal May 2022
Self-adaptive physics-informed neural networks journal February 2023
Self-adaptive loss balanced Physics-informed neural networks journal July 2022
Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre–Green–Naghdi equations journal March 2022
AT-PINN: Advanced time-marching physics-informed neural network for structural vibration analysis journal March 2024
nn-PINNs: Non-Newtonian physics-informed neural networks for complex fluid modeling journal January 2022
Physics-informed neural networks for solving Reynolds-averaged Navier–Stokes equations journal July 2022
Three-dimensional laminar flow using physics informed deep neural networks journal December 2023
Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows journal December 1998
Artificial neural networks for solving ordinary and partial differential equations journal January 1998
Continuous Chaos—Four Prototype Equations journal February 1979
Physics-Informed Neural Networks for Heat Transfer Problems journal April 2021
Some simple chaotic jerk functions journal June 1997
The Korteweg–deVries Equation: A Survey of Results journal July 1976
Physics-Informed Neural Networks with Hard Constraints for Inverse Design journal January 2021
Deterministic Nonperiodic Flow journal March 1963
Physics-informed neural networks for inverse problems in nano-optics and metamaterials journal January 2020

Figures / Tables (22)

Figure 1(p. 6)figure Figure 1
Figure 2(p. 9)figure Figure 2
Figure 3(p. 10)figure Figure 3
Table 1(p. 10)table Table 1
Figure 4(p. 14)figure Figure 4
Figure 5(p. 15)figure Figure 5
Table 2(p. 16)table Table 2
Table 3(p. 16)table Table 3
Table 4(p. 16)table Table 4
Figure 6(p. 18)figure Figure 6
Figure 7(p. 19)figure Figure 7
Figure 8(p. 20)figure Figure 8
Table 5(p. 20)table Table 5
Figure 9(p. 21)figure Figure 9
Table 6(p. 21)table Table 6
Figure 10(p. 23)figure Figure 10
Table 7(p. 23)table Table 7
Figure 11(p. 24)figure Figure 11
Table 8(p. 25)table Table 8
Figure 12(p. 26)figure Figure 12
Figure 13(p. 27)figure Figure 13
Table C.9(p. 30)table Table C.9

Similar Records

Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems
Journal Article · Fri Mar 18 00:00:00 EDT 2022 · Computer Methods in Applied Mechanics and Engineering · OSTI ID:1976976
Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics
Journal Article · Fri Jan 12 23:00:00 EST 2024 · Communications Physics · OSTI ID:2527398
Constrained or unconstrained? Neural-network-based equation discovery from data
Journal Article · Tue Dec 31 23:00:00 EST 2024 · Computer Methods in Applied Mechanics and Engineering · OSTI ID:2589832

Related Subjects

42 ENGINEERING
97 MATHEMATICS AND COMPUTING
causality
chaotic equations
deep learning
physics-informed neural networks (PINNs)
temporal continuity