Source-linked AI summary
Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks
Colby L. Wight, Jia Zhao
TL;DR
Accurate, efficient, and stable numerical solution of nonlinear, stiff phase-field equations remains challenging, and direct PINNs can fail when collocation points are randomly fixed. The paper improves PINNs with adaptive spatial and temporal sampling and reports accurate solutions, including a best relative l2 error of 9.51e−3 for a Cahn-Hilliard trial.
Problem
Nonlinearities and stiff terms make accurate, efficient, and stable numerical approximation of Allen-Cahn and Cahn-Hilliard phase-field equations challenging, while original PINNs use randomly selected points fixed during training.
Method
The paper improves PINNs with adaptive collocation-point sampling in space and time, alongside training strategies such as mini-batching and weighted loss functions.
Results
Only methods using adaptive sampling in both space and time accurately learned the Cahn-Hilliard solutions, with the best reported relative l2 error equal to 9.51e−3.
Takeaways & Limitations
The improved PINN could readily be used for other difficult phase-field equations, and combining the presented techniques produced the best performance in the reported study.
Takeaways & Limitations
For γ2 = 4, the neural network failed to learn the larger curves and did not capture the sharp transition layer well.
Abstract
from arXiv · showhide
Phase field models, in particular, the Allen-Cahn type and Cahn-Hilliard type equations, have been widely used to investigate interfacial dynamic problems. Designing accurate, efficient, and stable numerical algorithms for solving the phase field models has been an active field for decades. In this paper, we focus on using the deep neural network to design an automatic numerical solver for the Allen-Cahn and Cahn-Hilliard equations by proposing an improved physics informed neural network (PINN). Though the PINN has been embraced to investigate many differential equation problems, we find a direct application of the PINN in solving phase-field equations won't provide accurate solutions in many cases. Thus, we propose various techniques that add to the approximation power of the PINN. As a major contribution of this paper, we propose to embrace the adaptive idea in both space and time and introduce various sampling strategies, such that we are able to improve the efficiency and accuracy of the PINN on solving phase field equations. In addition, the improved PINN has no restriction on the explicit form of the PDEs, making it applicable to a wider class of PDE problems, and shedding light on numerical approximations of other PDEs in general.
1 Introduction
The paper studies deep-neural-network solvers for Allen-Cahn and Cahn-Hilliard phase-field equations, whose nonlinear and stiff terms make accurate, efficient, and stable numerical approximation challenging. It improves PINNs through adaptive sampling and related training strategies.
- Phase-field models: Allen-Cahn and Cahn-Hilliard equations model non-conservative and conservative phase variables in phase separation and arise as gradient flows of free-energy functions.Their generality makes them useful for interfacial and multiphase problems.
- Research problem: The paper investigates deep neural networks as numerical approximation methods for phase-field equations with nonlinearities and stiff terms.The stated goal is to improve the capabilities of neural networks for solving the Allen-Cahn and Cahn-Hilliard equations.
- Neural-network foundation: Feed-forward neural networks represent nonlinear mappings through layered weights, biases, and activation functions, with parameters learned by minimizing a loss function.Backpropagation supplies gradients for optimization, while stochastic gradient descent can use batches of training data.
- Physics-informed neural networks: PINNs extend neural-network approaches to differential equations, but the original method randomly selects training points once and keeps them fixed throughout training.This sampling choice failed to address issues relevant to the paper’s phase-field problems.
- Proposed improvement: The paper’s major contribution is adaptive sampling across space and time to improve PINN accuracy and efficiency for phase-field equations.The authors present several strategies and numerical examples to justify the approach.
2 Improved Physics Informed Neural Networks
The paper improves PINNs for phase-field equations with weighted losses, mini-batching, and adaptive sampling in space and time. These strategies address inaccurate direct PINN solutions, moving interfaces, and sharp transitions that can prevent convergence.
- 2.1 Physics informed neural networks: PINNs use neural networks to approximate the solution and its PDE residual, with training based on initial, boundary, and interior collocation data.The u-network approximates the solution, while the f-network approximates the equation residual; a small combined loss indicates a good approximation.
- 2.2 Improving PINN approximation: Direct PINN application to phase-field equations can produce inaccurate solutions, motivating techniques that improve approximation capability.The difficulty is associated with phase-field behavior, including sharp transitions and moving interfaces.
- 2.2 Improving PINN approximation: Weighted losses emphasize learning earlier times because dissipative phase-field equations must be solved forward in time.If the solution is not learned accurately at an earlier time, later-time accuracy is difficult to obtain.
- 2.2 Improving PINN approximation: Mini-batching uses subsets of training data for gradient updates and can facilitate convergence when approximating phase-field equations.The paper contrasts this with the non-mini-batched training used in the original PINN and follow-up work.
- 2.3 Adaptive strategies: Adaptive spatial sampling periodically adds collocation points in high-error regions, using f-network error to target moving interfaces while retaining domain-wide accuracy.Initial Latin hypercube points remain, while newly sampled points focus on harder regions and can be added over multiple resampling iterations.
- 2.3 Adaptive strategies: Time adaptivity either expands one network’s learned time interval after error decreases or trains separate networks on successive intervals.The first approach retains earlier-time points; the second passes the previous network’s predicted endpoint values as the next network’s initial condition.
- 2.3 Adaptive strategies: Adaptive strategies in space and time improve baseline PINN accuracy for phase-field solutions with sharp transitions.The paper attributes this improvement to avoiding certain local minima or saddle points during training.
3 Numerical Results
The experiments evaluate PINN variants for Allen-Cahn and Cahn-Hilliard equations, showing that adaptive sampling improves accuracy and convergence, especially when adapting in both space and time.
- Experimental setup: The Allen-Cahn and Cahn-Hilliard tests use initial and boundary data, interior collocation points, and relative l2 error against high-accuracy classical solutions.The u-network approximates the solution, while the f-network approximates the PDE residual, ideally producing zero residual.
- Allen-Cahn results: Adaptive spatial resampling produced a relative l2 error of 2.33e−02 using 2,000 collocation points, compared with typical 10,000-point non-adaptive tests.The adaptive solution was much better, though it still did not exactly match the real solution at later time steps; resampling was essential for convergence.
- Allen-Cahn results: For sharper Allen-Cahn interfaces, spatial resampling converged for γ2=1 and γ2=2 but failed within a reasonable number of iterations for γ2=3 and γ2=4.The γ2=3 failure began before t=.35, with error enlarging by t=1; the γ2=4 network could not learn the larger curves.
- Adaptive time sampling: Time-adaptive sampling improved difficult Allen-Cahn cases, learning γ2=4 with relative l2 error 0.04 and producing accurate predictions across the domain.A second time-adaptive approach also improved predictions near t=1 and accurately captured drop shrinking and bubble merging dynamics.
4 Conclusion
The paper improves PINNs for increasingly complex Allen-Cahn and Cahn-Hilliard phase-field equations through adaptive sampling and related training strategies. Combining the presented techniques generally produced the best performance, while the improved PINN may also apply to other difficult phase-field equations.
- The paper introduces several strategies to improve the approximating capability of PINNs for phase-field equations.
- Although demonstrated on Allen-Cahn and Cahn-Hilliard equations, the improved PINN could readily solve other difficult phase-field equations.
- Adaptive space sampling uses f-network predictions to focus collocation points, while time sampling uses differential-equation knowledge to focus temporal regions.
- The loss function and f-network predictions can guide decisions about adding collocation points or narrowing the time domain.
- The best performance was obtained by combining the presented techniques, with mini-batching and weighted loss functions remaining useful alongside adaptive sampling.