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Discontinuity Computing using Physics-Informed Neural Network
Li Liu, Shengping Liu, Hui Xie, Fansheng Xiong, Tengchao Yu, Mengjuan Xiao, Lufeng Liu, Heng Yong
TL;DR
PINNs remain limited in capturing sharp discontinuities, with shock-interior transition-points creating a training difficulty. The paper introduces local equation weighting to focus PINN training on smooth regions, and reports sharp discontinuity capture with results similar to or better than WENO-Z in the Sod example.
Problem
PINNs are still weak at calculating discontinuities compared with traditional shock-capturing methods, while shock-interior transition-points can dominate training.
Source: §1 · pp. 1–2Method
PINN-WE applies a local positive weight to governing equations in compressed regions, reducing shock-region influence so training focuses on smooth regions.
Source: §1 · p. 2, §4 · p. 17Results
PINN-WE achieved similar or better results than WENO-Z in the Sod problem and captured 2D contact discontinuities more sharply.
Source: §3.2.1 · p. 7, §3.2.3 · p. 10Takeaways & Limitations
The paper presents PINN as a way to capture discontinuities, especially shock waves, in Euler equations.
Source: §4 · p. 17For Euler equations, PINN-WE can converge to a discontinuous but nonphysical weak solution, so effective entropy conditions remain needed.
Source: §4 · p. 17
Abstract
from arXiv · showhide
Simulating discontinuities is a long standing problem especially for shock waves with strong nonlinear feather. Despite being a promising method, the recently developed physics-informed neural network (PINN) is still weak for calculating discontinuities compared with traditional shock-capturing methods. In this paper, we intend to improve the shock-capturing ability of the PINN. The primary strategy of this work is to weaken the expression of the network near discontinuities by adding a gradient-weight into the governing equations locally at each residual point. This strategy allows the network to focus on training smooth parts of the solutions. Then, automatically affected by the compressible property near shock waves, a sharp discontinuity appears with wrong inside shock transition-points compressed into well-trained smooth regions as passive particles. We study the solutions of one-dimensional Burgers equation and one- and two-dimensional Euler equations. Compared with the traditional high-order WENO-Z method in numerical examples, the proposed method can substantially improve discontinuity computing.
1. Introduction
PINNs face difficulty capturing sharp, non-oscillatory discontinuities: transition-points inside shocks can dominate training even though the strong-form equations do not govern the shock interior. The paper introduces a weighted-equation strategy that reduces emphasis on compressed regions so training can focus on smooth regions.
1. Introduction
- The paper targets improved PINN shock-capturing, especially for complex problems with shock generation.The introduction notes that forward-problem applicability of the original PINN has been limited to simple cases such as tracking moving shocks.Source: §1 · pp. 1–2
- Transition-points inside shocks can dominate PINN loss, yet the strong-form equations do not govern the shock interior and neither increasing nor decreasing their gradients resolves the conflict.The resulting loss can also interfere with convergence in smooth regions.Source: §1 · p. 2
- Unlike WENO’s locally reduced order and added dissipation in discontinuous cells, PINN transition-point losses can influence convergence in other regions.The authors contrast this with WENO cells whose large errors do not affect accuracy order beyond the discontinuity.Source: §1 · p. 2
- The authors introduce a ‘retreat to advance’ strategy that weakens network expression in strong-compression regions so training focuses on smooth regions.They describe the compressing property near shocks as allowing a sharp discontinuity to emerge.Source: §1 · p. 2
2. Method
The method section sets up PINN residual and initial/boundary-condition losses, then uses Burgers equation to show why shock-interior transition-points hinder training. PINN-WE locally weights governing equations in compressed regions to reduce their influence while preserving the original equations’ solutions for positive weights.
2.1. PINN for conservative hyperbolic PDE
- PINN represents the solution with a neural network and trains it using governing-equation residuals at domain points plus initial and boundary-condition losses.The PDE derivatives are evaluated automatically.Source: §2.1 · pp. 2–3
2.2. Analysis of transition-points based on Burgers equation
- In Burgers equation, transition-points carry nearly all PDE function loss, and after 3,000 epochs traditional PINN struggles to reduce total loss effectively.Increasing or decreasing the transition-point gradient raises the loss; because total loss is global, these points also affect errors elsewhere.Source: §2.2 · p. 3
2.3. PINN-WE
- PINN-WE assigns a local positive, gradient-dependent weight to governing equations in highly compressible regions, weakening transition-point influence so training emphasizes smooth regions.The proposed mechanism relies on compression to move transition-points into smooth regions, where sharp discontinuities emerge; positive weights preserve the original equation’s solutions.Source: §2.3 · p. 5
3. Numerical examples
The numerical examples evaluate PINN-WE on Burgers and Euler problems, comparing its results with traditional high-order WENO-Z. Across the examples, the reported findings include sharper discontinuities and cases where limited training data constrain detailed structures.
3.2. Euler equations
- The Euler tests apply PINN to weighted equations directly and compare Sod, Lax, 2D Riemann, and moving-shock problems with WENO-Z.The comparison uses fifth-order WENO-Z for spatial discretization and third-order Runge–Kutta for time integration.Source: §3.2 · p. 7
3.2.1. Sod problem
- PINN-WE achieves similar or better results than WENO-Z on the Sod problem, with no transition-points inside the shock.During training, PINN straggles at the shock while PINN-WE avoids that region.Source: §3.2.1 · p. 7, Figure 4 · p. 8
3.2.2. Lax problem
- For the Lax problem, PINN-WE correctly captures shocks in comparison with the high-order WENO-Z method.The problem contains a strong shock and strong contact; training reached a total loss of 0.01 after 20,000 epochs.Source: §3.2.2 · p. 9
3.2.3. 2D Riemann problem
- At 400×400 evaluation, PINN-WE captures 2D contact discontinuities more sharply than WENO-Z, though its detailed middle-region structure is weaker with the available training data.The authors also report that contacts are captured nearly without transport and smoothing points.Source: §3.2.3 · p. 10, Figure 10 · p. 13, Figure 11 · p. 14
3.2.4. Transonic flow through circular cylinder
- PINN-WE produces sharp, similar but smoother results than WENO-Z for transonic flow, and clearly captures the vertex behind the cylinder.The authors note that hardware-limited collocation samples prevented further loss reduction for resolving small structures.Source: §3.2.4 · p. 10, Figure 12 · p. 15, Figure 13 · p. 15
4. Conclusions
The paper proposes a PINN approach for capturing discontinuities in Euler equations by adjusting network expression according to local physical features. It reports several unresolved challenges, including physical-solution convergence, shock-position accuracy, and limited detail resolution.
4. Conclusions
- The proposed PINN approach targets discontinuity capture, especially shock waves, in Euler equations.It introduces a positive gradient-dependent weight into the governing equations, constructed inversely to local physics compression measured through velocity divergence.Source: §4 · p. 17
- Small weights in shock regions shift training toward smooth regions, where physical compression supports the automatic appearance of discontinuities.The inside-shock transition-points move into smooth regions like passive particles.Source: §4 · p. 17
- The study focuses only on discontinuity capture, while many Euler-equation questions and problems in complex discontinuous simulations remain open.Source: §4 · p. 17
- Convergence to a real physical weak solution remains difficult; the Euler equations may yield a nonphysical weak solution despite fast convergence with WE.Source: §4 · p. 17
- Shock position may be inaccurate without exact conservation of mass, momentum, and total energy in the PINN model.Source: §4 · p. 17
- As a global method, PINN can simulate big structures suitably but may inadequately reflect detailed structures, so resolution needs improvement.The authors identify improved resolution as necessary to increase applicability to diverse problems.Source: §4 · p. 17