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PINN-LPCDD: Physics-Informed Neural Network with Learnable and Physically Consistent Domain Decomposition for Compressible Multi-Medium Flows
Weiheng Zeng, Hao Wang, Jiaxin Zhu, Kun Wang, Tiegang Liu, Chengliang Feng
TL;DR
PINNs face challenges applying sharp-interface methods to compressible multi-medium flows, including shock approximation and interface learning. PINN-LPCDD combines medium-specific networks with learnable interfaces and related treatments for discontinuities and variable scales; the reported tests show sharp contact capture and learned interface speeds consistent with exact values.
Problem
Few works have applied sharp-interface methods to PINN-based compressible flow solving, and shock discontinuities remain difficult for PINNs to approximate.
Source: §1 · pp. 2–3Method
PINN-LPCDD assigns sub-networks to separate media, learns interface motion through interface continuity loss, and uses scaling transformations and a shock-representation mechanism.
Source: Conclusion · p. 30Results
PINN-LPCDD reports markedly smaller contact-discontinuity errors than PINN-AV, diminishing numerical oscillations during training, and learned interface velocities that agree well with exact values.
Source: Conclusion · p. 31Takeaways & Limitations
On some 2D test cases, the method’s computational efficiency is close to that of traditional methods.
Source: §1 · p. 3The marker-point interface parameterization has difficulty with complex geometries such as large deformation, merging, and breaking.
Source: Conclusion · p. 31
Abstract
from arXiv · showhide
The physics-informed neural network (PINN) faces significant challenges when solving compressible multi-medium flows, including the inability to locate interface positions, poor approximation of discontinuities, and difficulty in handling high-pressure-ratio problems between media with large stiffness differences. To address these issues, we propose PINN with Learnable and Physically Consistent Domain Decomposition (PINN-LPCDD), a framework that incorporates the sharp interface method (SIM) into PINNs for solving compressible multi-medium flows. Firstly, we design separate sub-networks for each medium with learnable parameters to automatically learn interface positions, so that the domain decomposition is physically consistent, and the contact discontinuity at the interface is naturally sharp. Secondly, the Euler equations with artificial viscosity are solved to enhance robustness when handling problems involving shock waves. Lastly, the scaling invariance of the Euler equations is exploited to facilitate the solution of high pressure ratio problems. Numerical experiments on air-air, air-helium, air-water, and air-R22 problems demonstrate the acceptable accuracy of the proposed method in simulating multi-medium flows.
1 Introduction
Compressible multi-medium flows require accurate shock capture and interface treatment because material interfaces separate distinct equations of state while some flow variables jump and others remain continuous. The introduction presents PINN-LPCDD as a sharp-interface PINN framework addressing interface learning, discontinuity capture, and large pressure-ratio challenges.
Motivation and challenges
- Multi-medium interfaces separate regions with different equations of state; density, sound speed, and EOS parameters jump, while pressure and normal velocity remain continuous.Shock interactions with these interfaces produce complex wave patterns, requiring interface treatment consistent with the EOS discontinuity.Source: §1 · p. 2
Related approaches
- Existing multi-medium numerical approaches include diffuse-interface models and sharp-interface methods using front tracking or level sets with ghost-fluid conditions.Diffuse-interface methods regularize interfaces into smooth transition layers.Source: §1 · p. 2
Motivation and challenges
- The introduction identifies three PINN challenges: learning interface locations with physically consistent conditions, capturing shocks, and handling data-scale disparities from large pressure ratios.Prior domain-decomposition methods fix interface positions before training, while shock-discontinuity approximation remains an open problem.Source: §1 · pp. 2–3
Contributions
- PINN-LPCDD is presented as the first framework to incorporate the sharp interface method into PINNs for compressible multi-medium flows, learning physically meaningful material interfaces through continuity conditions.Unlike cPINN/XPINN, its interface positions are learned during training rather than fixed beforehand.Source: §1 · p. 3
- The framework uses artificial viscosity for shocks and learnable domain decomposition to keep contact discontinuities naturally sharp.The introduction reports numerical validation of both capturing schemes.Source: §1 · p. 3
- Scaling invariance of the Euler equations is used to preprocess physical variables, with experiments demonstrating the need for preprocessing when media have large stiffness differences.The air-water Riemann problem is cited as an example with a 104:1 pressure ratio, which creates a data-scale disparity for neural-network approximation.Source: §1 · p. 3
2 Preliminaries
The preliminaries set out the two-medium Euler formulation and the PINN paradigms used to solve it. They define the material parameters and interface continuity conditions, then identify full space–time regression as this paper’s chosen paradigm.
2 Preliminaries
- The preliminaries introduce the governing equations for two-medium compressible flows and distinguish the PINN paradigms by how they handle time.The paper adopts the full space–time regression paradigm.Source: §2 · p. 4, §2.2 · p. 5
2.1 Euler equations for two-medium flows
- The two-medium Euler formulation uses primitive variables as network unknowns and a stiffened-gas EOS, with Γ and B constant within each medium.Γ is the ratio of specific heats and B is the EOS stiffness parameter; interface conditions require continuous normal velocity.Source: §2.1 · pp. 4–5
2.2 Paradigms for solving Euler equations with PINNs
- Full space–time regression approximates primitive variables over space and time and penalizes PDE residuals; time-stepping operator learning instead learns a map from one time level to the next.The paper adopts the first paradigm.Source: §2.2 · p. 5
3 Methodology
PINN-LPCDD combines separate medium-specific networks with a learnable interface and physically imposed continuity conditions. Scaling invariance addresses large variable disparities, while artificial viscosity and network design target shocks and sharp interfaces.
3.1 Data preprocessing
- Scaling the Euler variables before solving and transforming them back addresses the ill-conditioned losses associated with large pressure disparities.The method uses a two-parameter scaling invariance; air-water Riemann problems can have a pressure ratio of 104:1.Source: §3.1 · p. 6
3.2 Interface parameterization
- Interface marker velocities are learned through physical continuity conditions, avoiding a fixed interface position and reducing the cost of learning a full level-set surrogate.For closed curves, endpoint markers are connected to define normals; the parameterization follows front tracking but learns marker velocities during training.Source: §3.2 · pp. 7–8
3.3 Network architecture
- Separate networks preserve genuine jumps across material interfaces, while Fourier features and localized RBF responses support high-frequency shocks and local refinement.The Fourier embedding is shared by the two sub-networks, whose fully connected weights remain independent; the RBF layer targets local solution structure.Source: §3.3 · pp. 8–10
3.4 Loss function
- Each medium's sub-network uses Euler residuals with artificial viscosity, and a jointly optimized interface loss enforces pressure and normal-velocity continuity.The total loss combines sub-network terms with interface continuity terms, optimizing both network and interface parameters.Source: §3.4 · pp. 10–12
4 Numerical Experiment
The numerical studies evaluate PINN-LPCDD on a series of problems using uniform computational-domain partitions and exact or reference solutions reserved for evaluation.
4 Numerical Experiment
- Exact solutions or HODG reference solutions are used only to evaluate predictions, not in the loss or training process.Source-free Riemann problems use an exact Riemann solver; other cases use the customized HODG platform.Source: §4 · p. 13
- Training grids use (Nt,Nx) = (64,512) for 1D problems and (Nt,Nx,Ny) = (16,128,64) for 2D problems.The points are generated by uniformly partitioning the computational domain.Source: §4 · p. 13
- Terminal-time accuracy is evaluated with mean squared error and relative L2 error for primitive variables.The reported MSE(avg) and RE(avg) are averages over the primitive variables.Source: §4 · p. 13
4.1 Air-air Riemann problems
PINN-LPCDD models the air-air contact discontinuity as an interface between two media and learns its propagation speed, with comparisons against PINN, PINN-AV, and PINN-WE. Across Sod, Lax, and double-shock tests, it captures discontinuities sharply, with results depending on the artificial-viscosity coefficient.
4.1.1 Sod problem
- In the Sod test, PINN-LPCDD captures the contact discontinuity sharply without smearing and achieves the smallest errors among the compared methods.Without artificial viscosity, vanilla PINN struggles to approximate the discontinuity.Source: §4.1.1 · p. 14, Table 1 · p. 14, Figure 2 · p. 15
4.1.2 Lax problem
- PINN-LPCDD captures the stronger contact discontinuity and centered rarefaction wave better than the other models in the Lax test, and achieves the smallest errors.PINN-AV has better overall approximation than vanilla PINN, although both approximate the plateau to the right of the contact poorly.Source: §4.1.2 · p. 16, Figure 3 · p. 16, Table 2 · p. 17
4.1.3 Double shock (DS) problem
- With ν = 2×10−4, PINN-LPCDD achieves its best accuracy in the double-shock test; ν = 2×10−3 oversmooths solutions, while ν = 2×10−5 has negligible effect.At ν = 2×10−4, it also approximates better than PINN-AV at the same viscosity; vanilla PINN shows oscillations near the contact discontinuity.Source: §4.1.3 · p. 17, Table 3 · p. 17, Figure 4 · p. 18
4.2 Air-helium Riemann problems
For air-helium problems, PINN-LPCDD learns interface positions, unlike cPINN/XPINN, which require interface equations a priori. Ablations show that Fourier features, the RBF output layer, and separate subnetworks contribute to approximation quality in the two pressure configurations.
4.2 Air-helium Riemann problems
- The air-helium tests use ablations of Fourier feature embedding, the RBF output layer, and separate subnetworks; ablated models are widened to keep parameter counts no lower than PINN-LPCDD.PINN-WE and PINN-AV are not directly applicable to these tests, and cPINN/XPINN would require an interface equation specified beforehand.Source: §4.2 · p. 20
4.2.1 Air with high pressure
- In the air-high-pressure test, PINN-LPCDD approximates the step to the right of the interface better than ablated variants, supporting the need for Fourier features, the RBF layer, and dual networks.The authors describe the task as distinguishing a contact discontinuity and centered rarefaction wave within less than one-fifth of the spatial domain.Source: §4.2.1 · p. 20, Table 4 · p. 20, Figure 6 · p. 21
4.2.2 Helium with high pressure
- In the helium-high-pressure test, removing Fourier features or the RBF layer prevents correct interface learning, while removing subnetworks smears the solution near the interface.The reported results support using all three designs together.Source: §4.2.2 · pp. 21–22, Figure 7 · p. 22, Table 5 · p. 23
4.3 Air-water Riemann problems
The air–water tests examine PINN-LPCDD under extreme pressure ratios in one- and two-dimensional explosion settings. Scaling choice strongly affects 1D accuracy, while both 2D formulations solve the explosion problem despite poorer interface-shape prediction in the direct 2D formulation.
4.3.1 1D plane flow
- λ = 10^2 gives the best 1D air–water performance; λ = 40 is more dissipative, while λ = 1 and µ = 10^4 fail to learn effectively.With λ = 10^2, the true density, velocity, and pressure fall within [0,1], and their gradients are reduced.Source: §4.3.1 · p. 23
- For the 1D plane-flow test, the study compares four scaling strategies to assess how scaling parameters affect the learned solution.Source: §4.3.1 · p. 23, Table 6 · p. 23
4.3.2 2D cylindrically symmetric flow (explosion under water)
- The 2D explosion evaluation includes a source-term 1D formulation and a direct 2D formulation, with time-varying interfaces represented by denser samples.The direct 2D test uses the upper half-plane and a reflecting boundary on its bottom surface.Source: §4.3.2 · pp. 24–25
- Both the 1D source-term and direct 2D approaches solve the underwater explosion, though the coarser 2D grid yields larger average relative error.The 1D interface is well predicted, whereas the direct 2D interface departs from a circular shape without significantly affecting final results.Source: §4.3.2 · p. 25, Table 7 · p. 25
4.4 Shock-interface interaction
PINN-LPCDD is tested on shock interactions with air–R22 interfaces in one and two dimensions. It produces accurate 1D solutions before and after impact, while the 2D results are less satisfactory and the learned 1D interface velocities agree well with reference values.
4.4 Shock-interface interaction
- PINN-LPCDD yields accurate solutions in both phases before and after the shock impacts the air–R22 interface in the 1D test.The impact generates a transmitted strong shock in R22 and a reflected weak shock in air.Source: §4.4 · p. 27
- The 2D shock–bubble results are less satisfactory than the 1D results, with slight errors in left-traveling shock positions and some smearing of the reflected air shock.The relatively large average relative error is mainly attributed to the velocity component v, whose values are close to zero across most of the domain.Source: §4.4 · p. 28, Table 8 · p. 28, Figure 13 · p. 29
- Learned interface velocities in the 1D test cases have relative L2 errors ranging from 10^-4 to 10^-2.The reported comparison is with the true interface velocity.Source: §4.4 · p. 28, Table 9 · p. 30
- The reported training time for the 2D shock-interface test case (4B) is already close to that of traditional methods.Training costs are summarized in Appendix B.Source: §4.4 · p. 30
Conclusion
PINN-LPCDD combines separate medium-specific networks, learnable interface parameterization, and scaled primitive variables to address discontinuities, moving interfaces, and large pressure ratios. Experiments support its use across multi-medium Riemann problems, while the authors identify limits in complex interfaces, long shock–bubble simulations, and manual scaling selection.
Conclusion
- The numerical validation spans single-medium and two-medium Riemann problems, including extreme-pressure-ratio air–water cases and two-dimensional extensions.Source: Conclusion · pp. 30–31
- The dual-network strategy avoids smearing contact discontinuities, with markedly smaller errors there than PINN-AV; any numerical oscillations gradually diminish during training.Source: Conclusion · p. 31
- Learnable interface parameterization automatically estimates interface speed without requiring an a priori interface equation, and learned velocities agree well with exact values.Source: Conclusion · p. 31
- For large-pressure-ratio problems, preprocessing is indispensable, and removing Fourier features or the dual-network design degrades accuracy.An appropriate scaling parameter, such as λ = 100 in air–water problems, is crucial for effective learning.Source: Conclusion · p. 31
- The marker-point interface representation has difficulty with large deformation, merging, and breaking, while shock–bubble simulations remain short in physical time and have room for accuracy improvement.Scaling parameters also require manual tuning based on physical-variable ranges, which requires prior problem knowledge.Source: Conclusion · p. 31
Appendix A: Computational form of the governing equations
Appendix A gives component-wise computational residual forms for three flow configurations. It notes that the direct cylindrical 1D equation can become singular at the origin, motivating a different computational form.
Appendix A: Computational form of the governing equations
- The appendix specifies residual forms for 1D plane flow, 1D cylindrically symmetric flow, and 2D plane flow.The residual includes the artificial-viscosity coefficient ν.Source: Appendix A · p. 32
- For 1D plane flow, the adopted computational form is reported to reduce training time and improve accuracy.Source: Appendix A · p. 32
- For 1D cylindrical symmetry, a specialized computational form avoids singularities at x = 0 that can arise from directly using Eq. (2.1).Source: Appendix A · p. 32
Appendix B: Training costs of PINN-LPCDD
Appendix B compares PINN-LPCDD's training costs with traditional high-precision methods, finding a 1D disadvantage but a narrower efficiency gap for short-time 2D shock–bubble simulation.
Appendix B: Training costs of PINN-LPCDD
- PINN-LPCDD's 1D training takes about 7–15 minutes, compared with 1.40 minutes for the DG-P2 reference solution of test case (3B).The 1D training cost is not competitive with traditional high-precision methods.Source: Appendix B · p. 33
- For 2D cases, PINN-LPCDD's computational cost is comparable to traditional high-precision methods, with fewer spatial sampling points and time steps.On test case (4B), training on a coarse space-time grid still required mini-batches because of GPU memory limits and took nearly two hours.Source: Appendix B · p. 33
- At the accuracy required for a reliable reference solution, traditional methods no longer hold a clear efficiency advantage over PINN-LPCDD.The 2D reference took 101 minutes; the model's single-GPU training time was about five minutes longer.Source: Appendix B · p. 33
- The efficiency comparison is limited to short-time shock–bubble simulation; long-time simulation efficiency remains uninvestigated.Table 10 summarizes runtime, training speed, and GPU memory usage across test cases; training epochs vary by test group.Source: Appendix B · p. 33, Table 10 · p. 33