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A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions

Michael Penwarden, Ameya D. Jagtap, Shandian Zhe, George Em Karniadakis, Robert M. Kirby

arXiv:2302.14227v2physics.comp-phcs.LG

TL;DR

Forward PINNs and temporal decompositions can encounter optimization challenges, including incorrect or trivial solutions. The paper classifies these challenges and proposes a unified causality framework with stacked-decomposition and window-sweeping; its reported methods reduce training time and improve scalability over unmodified PINNs on problems without training challenges.

  • Problem

    Training forward PINNs remains challenging, and temporal decompositions such as XPINNs can face related problems involving information propagation.

    Source: §1 · p. 2, §3.1.1 · p. 8

  • Method

    The paper proposes a unified causality framework that includes stacked-decomposition, transfer-learning strategies, and window-sweeping collocation-point segmentation.

    Source: §1 · p. 2, §4.1 · p. 11, §4.1.2 · p. 12

  • Results

    The proposed methods reduce training time and improve scalability over an unmodified PINN on problems without training challenges.

    Source: §6 · p. 21

  • Takeaways & Limitations

    The framework describes existing and new causality-enforcing PINN methods and offers adaptive propagation based on loss tolerances rather than fixed optimization iterations.

    Source: §6 · p. 21

    Information propagation is not fully understood and depends on factors such as the optimizer and sampling method, which are not all studied here.

    Source: §3.1.2 · p. 10

Abstract

from arXiv · show

Physics-informed neural networks (PINNs) as a means of solving partial differential equations (PDE) have garnered much attention in the Computational Science and Engineering (CS&E) world. However, a recent topic of interest is exploring various training (i.e., optimization) challenges - in particular, arriving at poor local minima in the optimization landscape results in a PINN approximation giving an inferior, and sometimes trivial, solution when solving forward time-dependent PDEs with no data. This problem is also found in, and in some sense more difficult, with domain decomposition strategies such as temporal decomposition using XPINNs. We furnish examples and explanations for different training challenges, their cause, and how they relate to information propagation and temporal decomposition. We then propose a new stacked-decomposition method that bridges the gap between time-marching PINNs and XPINNs. We also introduce significant computational speed-ups by using transfer learning concepts to initialize subnetworks in the domain and loss tolerance-based propagation for the subdomains. Finally, we formulate a new time-sweeping collocation point algorithm inspired by the previous PINNs causality literature, which our framework can still describe, and provides a significant computational speed-up via reduced-cost collocation point segmentation. The proposed methods form our unified framework, which overcomes training challenges in PINNs and XPINNs for time-dependent PDEs by respecting the causality in multiple forms and improving scalability by limiting the computation required per optimization iteration. Finally, we provide numerical results for these methods on baseline PDE problems for which unmodified PINNs and XPINNs struggle to train.

1. Introduction

Continuous-time PINNs retain discretization flexibility, but their optimization remains difficult for forward problems; the paper studies training challenges and proposes a unified framework for causality-enforcing methods.

1. Introduction

  • Continuous-time PINNs are motivated by discretization flexibility, while the proposed direction seeks to retain that flexibility and gain properties of discrete-time approaches.The authors identify optimization as a major source of difficulty and note issues such as poor sampling, unequal loss weights, and unsuitable optimization schemes.Source: §1 · pp. 1–2
  • Forward continuous-time PINN training is the paper’s focus because it remains challenging, while inverse problems do not pose the same training issues.The authors describe information propagation as central to forward-problem challenges and leave inverse-problem methods for future work.Source: §1 · p. 2
  • The paper classifies PINN training challenges and relates them to information propagation and temporal decomposition, including challenges that also arise in XPINNs.It presents a unified, backward-compatible framework bridging time-marching and XPINNs, with collocation-point partitioning and transfer-learning concepts to reduce computational cost.Source: §1 · p. 2

2. Background

The background presents PINNs as neural networks trained to satisfy data and PDE-residual losses, reviews temporal causality methods, and classifies causality enforcement to identify gaps addressed by the paper.

2.1. Physics-Informed Neural Networks (PINNs)

  • PINNs optimize a neural-network solution using a loss that balances initial/boundary data fit with PDE-residual enforcement at collocation points.Additional loss terms, including interface terms for domain decomposition, can be used in PINN variants.Source: §2.1 · p. 3

2.2. Related work

  • Adaptive time-sampling adds collocation subsets over time, time-marching trains slabs sequentially, and backward-compatibility preserves prior-time predictions as a data term.These approaches enforce or stage temporal information at slab scale; adaptive sampling can reduce active collocation work before all slabs are added.Source: §2.2 · p. 4, Figure 1 · p. 4
  • Causal weighting applies a mask based on cumulative residuals from earlier times, but its sensitivity to ϵ can require costly repeated passes over the domain.XPINNs instead train stitched subnetworks in parallel and are not causal by default; their interface conditions differ from sequential time-marching.Source: §2.2 · pp. 4–5

2.3. Causality classification

  • The paper distinguishes hard causality, which cannot be violated, from soft causality, which optimization only predisposes a network to obey.Table 1 classifies prior causality methods using this distinction.Source: §2.3 · p. 5, Table 1 · p. 5
  • The identified gap is hard causality at sampling scale and methods combining hard and soft constraints; stacked-decomposition and window-sweeping are proposed to address it.The framework connects time-marching with XPINNs, while window-sweeping limits computed collocation residuals and enforces causality continuously.Source: §2.3 · p. 5

3. Training Challenges: PINNs and their temporal decompositions

The section classifies PINN training challenges through information propagation and shows how temporal decomposition can intensify them when subnetworks train concurrently. It contrasts these issues with time-marching, which limits optimization to preserve causal progression.

3.1.1. Types of Training Challenges

  • Long-time periodic convection can converge toward zero because the PDE residual admits constants and initial-condition information has not yet propagated across the domain.The long-time case’s residual is zero at later times; time-marching can alleviate the issue by preventing optimization from converging to that later zero-solution.Source: §3.1.1 · pp. 6–7, Figure 3 · p. 7
  • Too few residual collocation points cause convection’s initial-condition features to stop abruptly, leaving a constant solution across the remaining domain.The paper reports consistently solving the problem with more points and identifies increased and adaptive sampling as approaches to investigate.Source: §3.1.1 · pp. 7–8
  • Allen–Cahn can converge quickly to an incorrect solution, a distinct failure from zero-solution stagnation that arises when strong causal enforcement is needed.The paper notes that long-time KdV exhibits incorrect propagation rather than the zero-solution challenge.Source: §3.1.1 · p. 8

3.1.2. Temporal Decomposition Challenges

  • Concurrent XPINN subnetworks can stall at the first interface in the zero-solution minimum, and more training iterations do not resolve the propagation problem.Later subnetworks lack direct access to the initial condition; causality enforcement is needed to alleviate the issue.Source: §3.1.2 · p. 9
  • For temporal decomposition, solution continuity with Dirichlet boundary conditions obtains the correct solution in the reported comparison, whereas standard XPINN interface conditions still impede propagation.Weak rather than exact periodic-boundary enforcement gives the same correct and incorrect outcomes; the paper reports that changing boundary-condition type, such as to Dirichlet, changes the result.Source: Figure 4 · p. 9, §3.1.2 · p. 10
  • Time-marching removes later-time collocation points from optimization, helping avoid trivial zero-solutions; information propagation also depends on aspects such as optimizer and sampling that the study does not cover fully.The paper cautions that propagation is not fully understood and notes that complex characteristics, such as in 3D Euler, make some methods difficult.Source: §3.1.2 · p. 10

4. Unified Causality-Enforcing Framework

The framework unifies causal enforcement across temporal subdomains and collocation sampling, combining soft and hard constraints with transfer-learning options.

4. Unified Causality-Enforcing Framework

  • The framework combines soft and hard causal constraints at both time-slab and sampling scales, with transfer learning as an additional way to improve temporal decomposition.It is proposed to address decomposition challenges and unify previous causal strategies.Source: §4 · p. 10

4.1. Stacked-decomposition

  • Stacked-decomposition spans time-marching (dS = 1) to traditional XPINNs (dS = n), with causal warm-up delaying later slabs until the preceding slab converges.Intermediate dS values offer a middle ground when full-domain training is expensive but prior-domain information remains useful.Source: §4.1 · p. 11, Figure 5 · p. 11

4.1.1. Interface Conditions

  • For first-order time problems, adjacent slabs use solution continuity; residual continuity is unnecessary for time decomposition but may be included if the resulting problem remains well-posed.The same continuity condition remains well-posed for irregularly shaped time-slabs.Source: §4.1.1 · pp. 11–12

4.1.2. Transfer learning

  • Initializing a subnetwork from its predecessor makes the interface or initial-condition loss zero, but freezing layers can inhibit fitting as Allen-Cahn spatial scales sharpen over time.Fine-tuning initializes from prior parameters while keeping the full network trainable; Figure 6 also shows parameter distributions staying nearly constant despite the basis change.Source: §4.1.2 · pp. 12–13, Figure 6 · p. 13

4.2. Window-sweeping collocation points

  • Window-sweeping moves a soft weight mask across one PINN's time collocation points; backward-compatibility points impose hard causality, while null-set cutoffs avoid costly future residuals until useful.The method's kernels can recover causal weighting, and uniform weights with width dt recover bc-PINNs; dt tolerance controls when the point-set bounds move.Source: §4.2 · pp. 13–14, Figure 7 · p. 13, Table 3 · p. 14

4.3. User Settings

  • The procedure exposes settings for decomposition, interfaces, transfer learning, and window-sweeping; a loss-change tolerance triggers subdomain addition, reducing unnecessary fixed-iteration training.The tolerance also reduces iteration-count tuning, since too few iterations can produce an incorrect solution and too many are expensive.Source: §4.3 · p. 14

5. Numerical Experiments

The experiments compare causal temporal-decomposition and window-sweeping settings on convection, Allen–Cahn, and Korteweg–de Vries problems, emphasizing accuracy, training cost, and practical trade-offs.

5. Numerical Experiments

  • Experiments use 10,000 LHS residual collocation points, 200 uniformly spaced boundary and initial/interface points, and 50-neuron, four-hidden-layer networks; lower loss tolerances may improve accuracy but require more training time, and reported hyperparameter choices are not exhaustive.Ground-truth solutions use a spectral Fourier discretization with 512 modes and ETDRK4 time stepping at 10^-5.Source: §5 · p. 15

5.1. Convection equation

  • On convection, stacked-decomposition with fine tuning can outperform the standard PINN in both accuracy and cost, while causal XPINN fine tuning can recover the correct solution at high computational cost.This holds even with periodic boundaries and XPINN interfaces, settings identified as unfavorable for training.Source: §5.1 · pp. 16–17, Table 5 · p. 16
  • For convection, increasing the number of temporal subdomains reduces error when Adam alone stalls, but beyond a problem-specific ideal it adds runtime without benefit.With 500 Adam warm-up iterations followed by L-BFGS, error stays relatively constant across subdomain counts; decomposition can still improve training time when all methods converge.Source: §5.1 · p. 17, Figure 9 · p. 17

5.2. Allen-Cahn equation

  • For Allen–Cahn, smaller time slabs have faster-decaying temporal-snapshot singular values and empirically train faster; the error-function window-sweeping kernel shows an even faster drop-off.The authors present this as a potential route to reduced training times, not a definitive convergence estimate.Source: §5.2 · p. 18, Figure 10 · p. 18
  • On Allen–Cahn, backward-compatibility sets substantially reduce training time versus repeatedly training on prior point sets, without adverse accuracy effects.The reported loss-tolerance propagation uses 10^-7; the authors report almost identical relative L2 errors to bc-PINNs with around 12,000 total iterations versus hundreds of thousands.Source: Table 6 · p. 18, §5.2 · p. 19
  • For Allen–Cahn, M=10 better resolves end-time discontinuities than M=1, but errors shift location and overall accuracies are similar at the reported stopping tolerances.M=10 has low point-wise error near the discontinuities but higher error in the smoother x=0 region; M=1 struggles at the discontinuities.Source: §5.2 · p. 19, Figure 11 · p. 19

5.3. Korteweg–de Vries equation

  • For short-time KdV, stacked-decomposition and window-sweeping separately improve accuracy and reduce training time; combining them adds overhead without benefit in this setting.The authors attribute the lack of benefit to the methods separately reaching the accuracy bound for the network size and tolerances.Source: §5.3 · p. 20, Figure 12 · p. 20, Table 7 · p. 20
  • For long-time KdV, combining stacked-decomposition and window-sweeping improves accuracy while keeping training time low relative to an unmodified PINN.With window width and dt both 0.5, window-sweeping alone required 500 Adam iterations at each propagation step to stabilize L-BFGS; this added training time.Source: §5.3 · pp. 20–21, Figure 13 · p. 21

6. Summary

The paper presents a unified framework for causality-enforcing PINN methods and reports improved scalability through adaptive loss-tolerance propagation. It also identifies second-order-in-time PDEs as future work.

6. Summary

  • The framework combines existing and new causality-enforcing methods, including stacked-decomposition and window-sweeping, to address training challenges in PINNs and temporal decompositions.It also relates these challenges to information propagation and introduces adaptive propagation based on changes in loss tolerance rather than fixed iteration counts.Source: §6 · p. 21
  • On problems without training challenges, adaptive propagation reduces training time and improves scalability over an unmodified PINN.The paper also investigates choices such as parameter transfer and boundary-condition enforcement to guide model decisions.Source: §6 · p. 21
  • Second-order-in-time problems, including wave and Boussinesq equations, remain future work because their information-propagation considerations differ from the standard forward-time causality setup.The authors also identify cases with only zeroth-order information at initial and final times as future work.Source: §6 · p. 21

Appendix A. Symbols and Notations

The notation appendix collects the paper’s symbols for the PINN solution and residual, temporal decompositions, collocation sets, and training choices.

Appendix A. Symbols and Notations

  • u(·) denotes the PDE solution, uθ(·) the PINN prediction, θ the learnable parameters, and R(·) the PDE residual.These symbols distinguish the target solution from its network approximation and residual.Source: Appendix A · p. 22
  • s-d means stacked-decomposition, w-s means window-sweeping, FT means fine tuning, and TL means transfer learning.Source: Appendix A · p. 22
  • Ni, Nr, Nic, and Nb denote interface, residual-collocation, initial-condition, and boundary-point counts, respectively.Source: Appendix A · p. 22
  • dS is the number of subnetworks trained at once, n the number of time slabs, and M the Fourier-feature encoding order.Source: Appendix A · p. 22

Appendix B.1. Convection: Fine Tuning vs. Transfer Learning

In the final time-slab network, fine tuning keeps parameters near [-1, 1], while transfer learning with frozen layers drives later-layer parameters to more extreme values.

Appendix B.1. Convection: Fine Tuning vs. Transfer Learning

  • Freezing the first two layers during transfer learning makes the final three layers over-adjust, with parameters spanning [-4, 4] rather than about [-1, 1] under fine tuning.The authors connect this compensation to reduced network expressivity and longer training times for transfer learning in temporal PINN decomposition.Source: Appendix B.1 · p. 22, Figure B.14 · p. 23
  • The parameter-distribution explanation is explicitly limited to temporal decomposition with PINNs, not transfer-learning trade-offs in other applications.Source: Appendix B.1 · p. 22

Appendix B.2. Allen-Cahn: Training Dynamics of Stacked-Decomposition & Window-Sweeping

Figure B.15 reports training loss and end-of-training full-domain PDE residual for stacked-decomposition and window-sweeping on Allen-Cahn. The dynamics contrast how the methods retain initial-condition information and handle interfaces.

Appendix B.2. Allen-Cahn: Training Dynamics of Stacked-Decomposition & Window-Sweeping

  • Stacked-decomposition can eventually omit the initial condition from loss minimization unless n = dS, so information is carried through later subdomains and interfaces.Window-sweeping always includes the initial condition during optimization.Source: Appendix B.2 · p. 23
  • Window-sweeping uses one network and keeps residuals and predictions continuous, whereas stacked-decomposition can show interface discontinuities.Multiple networks have more expressivity when training challenges are absent and interfaces are respected, which can yield smaller residuals, especially near the end of time.Source: Appendix B.2 · p. 23
  • Figure B.15 plots loss over training epochs and the full-domain PDE residual at training end for both methods.The displayed settings recover time-marching with fine tuning for stacked-decomposition and bc-PINNs for window-sweeping.Source: Appendix B.2 · p. 23, Figure B.15 · p. 24

Appendix B.3. Allen-Cahn: Causal Weights

The Allen-Cahn causal-weights results describe a loss-tolerance propagation setup, its adaptation to LHS sampling, and the resulting accuracy and training-time qualifications. The appendix does not claim which window-sweeping method is best in the extreme-training limit.

Appendix B.3. Allen-Cahn: Causal Weights

  • For Allen-Cahn, 10^-7 was used as the loss tolerance; a lower tolerance increased training time without improving accuracy, while a higher tolerance reduced accuracy.The same tolerance was used across settings to compare accuracy and training time fairly.Source: Appendix B.3 · p. 24
  • For smaller ϵ, causal weights do not enforce causality strongly enough to overcome the training challenge; the five-sweep cascading-ϵ approach is outside this study’s scope.The authors note that window-sweeping can also be used for multiple sweeps.Source: Appendix B.3 · p. 24
  • The causal-weight loss is formed from the mean of snapshot-wise mean squared errors, so the implementation is divided into time snapshots rather than weighting all residual points separately.For LHS, the authors order the 10,000 points by time into 100 sets of 100 to adapt the scheme while retaining spatial correlation when sampling is dense enough.Source: Appendix B.3 · pp. 24–25
  • LHS causal-weight window-sweeping showed no adverse performance effect from grouping points by time, and null-set segmentation reduced training time by skipping negligible-weight future residuals.The 10,000 points were ordered in time and divided into 100 sets of 100; adding future sets when miniwi > 0.05 enabled the segmentation.Source: Appendix B.3 · p. 25
  • The reported setting did not reach the 10^-3 relative L2 errors reported in [25]; the authors cite the looser δ termination and differences in iterations and network parameters as factors.They state that the original accuracy likely relied largely on 10–100× more iterations and 10–100× more network parameters θ, among other modifications.Source: Appendix B.3 · p. 25
  • All reported window-sweeping kernels achieve comparable results under the chosen settings, and the study makes no claim about the best accuracy or cost in the extreme-training limit.Table B.9 reports relative L2 error and training time for different window-sweeping settings, generally using M = 10.Source: Appendix B.3 · p. 25, Table B.9 · p. 25

Appendix B.4. KdV (long-time) PINN Prediction

The PINN solves the long-time KdV problem on T = [0, 1] but fails on T = [0, 5]. The failure is identified as incorrect information propagation rather than a zero-solution challenge.

Appendix B.4. KdV (long-time) PINN Prediction

  • The PINN solves KdV for T = [0, 1] but fails for T = [0, 5].Figure B.16 is presented as representative of the incorrect-propagation training challenge.Source: Appendix B.4 · p. 25, Figure B.16 · p. 25
  • The long-time KdV failure is classified as incorrect propagation, like the Allen-Cahn challenge, rather than the zero-solution issue in long-time Convection.The authors attribute it to increased training difficulty over a larger temporal domain.Source: Appendix B.4 · p. 25
  • The authors suggest periodic conditions may extend Convection’s traveling wave, a feature absent in KdV, while larger KdV time domains manifest as incorrect information propagation.Source: Appendix B.4 · p. 25

Appendix C. Fourier Feature encoding (C∞periodic conditions)

The appendix describes Fourier-feature encoding of the spatial input to exactly enforce C∞ periodic boundary conditions. It relates the frequency choice M to the frequency components the network can represent.

Appendix C. Fourier Feature encoding (C∞periodic conditions)

  • Fourier-feature encoding is applied to the network’s spatial input to exactly enforce C∞ periodic boundary conditions.Source: Appendix C · p. 26
  • A higher M yields higher-frequency output components after nonlinear activations, which may help with high-frequency PDE solutions such as Allen-Cahn.M is a non-negative integer representing the sinusoidal input frequency.Source: Appendix C · p. 26
  • All choices of M are shown to be C∞ periodic in Lemma 2.1 of.Source: Appendix C · p. 26
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