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Efficient and Accurate Lift-and-Decoupling Generalized Transferable Neural Networks for Elliptic and Moving Interface Problems

Tao Cheng, Junqing Jia, Lili Ju, Xiaoping Zhang

arXiv:2610.02891v1math.NAcs.NA

TL;DR

Interface problems involve solution or flux discontinuities that challenge approximation, and conventional lifting can create redundant representations. LD-GTransNet decouples spatial coordinates and an auxiliary variable in parallel subnetworks within a fixed-parameter GTransNet framework. The authors report high-order accuracy with significantly smaller linear systems in two- and three-dimensional experiments.

  • Problem

    Interface discontinuities can produce low-regularity or singular solutions, while conventional lifting can increase input dimensionality and cause representation non-uniqueness.

    Source: §1 · pp. 1, 3

  • Method

    LD-GTransNet processes physical coordinates and an auxiliary variable in separate subnetworks, combining their features within a fixed-hidden-parameter framework.

    Source: §1 · p. 3, §3.1 · p. 7

  • Results

    Numerical experiments in two and three dimensions show high-order accuracy while using significantly smaller linear systems.

    Source: §5 · p. 25

  • Takeaways & Limitations

    The framework is reported as applicable to elliptic equations, Stokes flow, anisotropic diffusion–reaction systems, elasticity problems, and moving interfaces.

    Source: §5 · p. 25

    The penalty-parameter selection is an empirically motivated heuristic, and rigorous analysis of its convergence properties remains to be investigated.

    Source: §3.1 · p. 10

Abstract

from arXiv · show

In this paper, we propose a highly efficient and accurate method, called ``lift-and-decoupling generalized transferable neural network" (LD-GTransNet), for numerical solution of elliptic and moving interface problems. Our method makes use of the multi-layer GTransNet approach and incorporates a novel {lift-and-decoupling} strategy, in which the hidden-layer neuron parameters are preset and the physical coordinates and the added auxiliary variable are processed by separate subnetworks. Such design enables a unified global approximation of piecewise smooth solutions without explicit domain decomposition, while preserving network expressivity and avoiding the redundancy inherent in conventional {lift} approaches. The resulting formulation reduces to a linear least-squares problem, thereby eliminating the need of nonlinear training and ensuring the high computational efficiency and accuracy. Extensive numerical experiments on benchmark interface problems in two and three dimensions demonstrate that, compared with existing neural network-based and traditional numerical methods, our proposed LD-GTransNet consistently delivers superior performance, particularly for problems with high-contrast coefficients and complex interface geometries.

1. Introduction

Interface problems create discontinuities that challenge numerical approximation, while mesh-based methods face geometric and interface-treatment burdens. LD-GTransNet combines GTransNet with a decoupled lifting strategy to represent piecewise-smooth solutions globally without domain decomposition or redundant lifted bases.

1. Introduction

  • Interfaces can induce discontinuities in solutions or fluxes and low regularity or singular behavior, making numerical approximation challenging.The problems arise in applications involving heterogeneous media and phase transitions.Source: §1 · p. 1
  • Fitted methods depend on high-quality interface-conforming meshes, while unfitted methods can require difficult cut-cell treatments and accurate geometric reconstruction.These challenges can reduce robustness and accuracy for interfaces with large curvature or intricate topology.Source: §1 · pp. 1–2
  • Meshfree methods avoid explicit mesh construction, and neural networks offer approximation flexibility for high-dimensional and irregular problems.Kernel-based methods and generalized finite differences are also identified as promising meshfree approaches.Source: §1 · p. 2
  • Fully parameterized neural PDE methods can be costly to train and difficult to optimize, while fixed-neuron methods still struggle to represent sharp jumps and incorporate interface conditions.Domain decomposition and input augmentation can address interfaces but add network or training complexity.Source: §1 · p. 2
  • Conventional lifting avoids explicit domain decomposition but concatenating physical and auxiliary variables increases input dimension and creates representation non-uniqueness.LD-GTransNet processes these variables in parallel subnetworks, using their Hadamard product to select spatial bases by subdomain label and remove redundancy while retaining approximation capability.Source: §1 · p. 3

2. Generalized Transferable Neural Networks for PDEs

TransNet uses predetermined hidden-layer parameters to form neural basis functions for PDE approximation. GTransNet extends this approach to multiple hidden layers, with predetermined hidden transformations and trainable output coefficients.

2.1. Transferable Neural networks

  • TransNet constructs a neural feature space from activated hidden-layer neurons and represents the PDE solution as a linear combination of those basis functions.Its hidden parameters are predetermined, with neuron directions and offsets sampled to provide uniform expressivity across the domain.Source: §2.1 · p. 4

2.2. Generalized transferable neural networks

  • GTransNet extends TransNet to L ≥ 2 hidden layers, retaining predetermined hidden parameters and using variance-controlled weights without biases in subsequent layers.The first layer inherits TransNet's sampling scheme with centrally symmetric neuron biases.Source: §2.2 · p. 5
  • The final-layer features form the basis of GTransNet for solution approximation, and the network output is their linear combination, parameterized by trainable coefficients α.If L = 1, GTransNet reduces to the original TransNet.Source: §2.2 · p. 5

3. Lift-and-Decoupling GTransNets for Elliptic and Moving Interface Problems

LD-GTransNet extends GTransNet to piecewise-smooth elliptic and moving-interface problems by lifting an auxiliary regional variable and processing it separately from spatial coordinates. For elliptic problems, this separates interior PDE and interface-jump residuals into a linear least-squares system; unknown moving interfaces are handled by alternating linear solves.

3.1. Elliptic interface problems

  • Separate spatial and auxiliary subnetworks combine their features by a Hadamard product, using a piecewise-constant regional indicator to gate spatial features.The spatial subnetwork represents smooth structures across the domain, while the auxiliary subnetwork remains fixed during spatial differentiation.Source: §3.1 · p. 7
  • Interior PDE and interface-jump residuals occupy independent least-squares block rows, with jumps encoded by differences between auxiliary basis functions.Row scaling balances residual magnitudes; its penalty-parameter selection is an empirically motivated heuristic whose convergence properties remain unanalysed.Source: §3.1 · pp. 7, 10
  • Without interfaces, the auxiliary differences vanish and the system reduces exactly to the original GTransNet least-squares formulation.Source: §3.1 · p. 10

3.2. Moving interface problems

  • For known interface motion, treating time as an input makes the moving-interface problem a static case in space-time.With unknown motion coupled to the solution, Gauss-Seidel block iteration alternates linear least-squares solves for the solution and interface parameters.Source: §3.2 · pp. 11–12
  • The moving-interface formulation represents an unknown two-phase interface as the zero level set of a function governed by a possibly nonlinear condition coupled to the subdomain PDEs and jump conditions.Source: §3.2 · p. 11
  • For a two-dimensional Stefan example, the interface is parameterized as x = s(y, t), and s may be represented directly by a TransNet.The framework is described for a generic two-phase moving-interface problem, with extension to multiple interfaces and phases stated to be straightforward.Source: §3.2 · pp. 11–13

4. Numerical Experiments

Experiments in two and three dimensions assess LD-GTransNet across diffusion, Stokes, elasticity, and moving-interface problems, using relative L2 and L∞ errors and repeated runs.

4. Numerical Experiments

  • Tests span two- and three-dimensional diffusion, Stokes, elasticity, and moving-interface problems, including multiple interfaces and varied geometries.The set includes isotropic diffusion, Stokes flow with a circular interface, anisotropic diffusion-reaction, and elasticity with an elliptic interface.Source: §4 · p. 13
  • Relative L2 and L∞ errors are used to evaluate numerical accuracy.The errors are calculated against the corresponding exact solution.Source: §4 · p. 13
  • Each experiment is repeated 10 times, with least-squares problems solved using SciPy’s scipy.linalg.lstsq API.Testing uses O(10^4) points for 2D examples and O(10^5) points for 3D examples.Source: §4 · p. 13

4.1. Tests for elliptic interface problems

The elliptic-interface tests examine LD-GTransNet on multiple-interface diffusion, Stokes, anisotropic diffusion-reaction, and 3D elasticity problems. Across these examples, the reported results show high accuracy and favorable comparisons with the selected neural-network and numerical baselines.

4.1.1. A 2D isotropic diffusion multiple-interface equation

  • LD-GTransNet achieves relative L2 errors of O(10^-12) with two or three hidden layers; two layers provide a comparable-accuracy, lower-cost trade-off.The authors use the two-hidden-layer configuration for subsequent results.Source: §4.1.1 · pp. 14–15, Figure 4 · p. 15

4.1.2. A 2D Stokes interface problem

  • With 4,800 degrees of freedom, LD-GTransNet improves accuracy by two to four orders of magnitude over RFM and DC-RFM across viscosity contrasts from 10 to 10^3.The pressure error degrades only slightly as viscosity contrast increases.Source: Table 2 · p. 17, §4.1.2 · p. 18

4.1.3. A 2D anisotropic diffusion-reaction multiple-interface problem

  • At N = 1800, relative L∞ and L2 errors reach O(10^-11) and O(10^-12), respectively, and LD-GTransNet substantially outperforms DCSNN in numerical accuracy.The authors contrast LD-GTransNet's linear least-squares solve with DCSNN's nonlinear Levenberg-Marquardt optimization, which incurs higher computational cost.Source: Figure 8 · p. 19, §4.1.3 · p. 20, Table 3 · p. 20

4.1.4. A 3D elasticity interface problem

  • For 3D elasticity with high-contrast Lamé parameters, LD-GTransNet performs better than high-order NIPFEM in number of constraints, degrees of freedom, and L2 error.The reported comparison is for (ν1, µ1, ν2, µ2) = (1, 1000, 1, 1000).Source: Figure 10 · p. 21, §4.1.4 · p. 22, Table 4 · p. 22

4.2. Tests for moving interface problems

The section tests LD-GTransNet on an evolving circular interface and a two-phase Stefan problem. It reports lower errors than the respective neural-network baselines in both settings.

4.2.1. A 2D moving interface problem

  • The test uses a circular interface whose radius evolves according to x^2 + y^2 = 0.5 + 0.3t.The diffusion coefficient is κ1 inside the circle and κ2 outside; the terminal time is T = 1.Source: §4.2.1 · p. 23
  • For the moving-circle test, time is treated as an additional spatial coordinate, turning the moving interface into a fixed hypersurface in lifted space.Collocation points are sampled once over the space-time domain rather than regenerated at individual time steps.Source: §4.2.1 · p. 23
  • Four to five orders of magnitude higher accuracy than INN with the same number of constraints.LD-GTransNet eliminates nonlinear optimization, while INN requires gradient-based training.Source: Table 5 · p. 23, §4.2.1 · p. 24

4.2.2. A 2D two-phase Stefan problem

  • For the Stefan test, the interface is x = s(y, t), separating regions with diffusion coefficients κ1 = 2 and κ2 = 1.The domain is [0, 2] × [0, 1] and the terminal time is T = 1.Source: §4.2.2 · p. 24
  • An L2 error three orders of magnitude lower than PIELM with the same number of constraints and degrees of freedom.The comparison concerns the two-phase Stefan problem.Source: Table 6 · p. 25, §4.2.2 · p. 25

5. Conclusion

The paper proposes LD-GTransNet for PDEs with interface discontinuities. Its fixed hidden-layer parameters and decoupled variables yield a global representation and a linear least-squares formulation; experiments report high-order accuracy and smaller linear systems, especially for high-contrast coefficients or complex, evolving interfaces.

5. Conclusion

  • LD-GTransNet decouples physical and auxiliary variables to represent piecewise smooth solutions globally without domain decomposition or the redundancy issue.The formulation uses fixed hidden-layer parameters and reduces to linear least squares, eliminating nonlinear optimization.Source: §5 · p. 25
  • Experiments in two and three dimensions show high-order accuracy with significantly smaller linear systems than existing methods.Advantages are especially pronounced for high-contrast coefficients or complex, evolving interfaces.Source: §5 · p. 25
  • The framework is applicable to elliptic equations, Stokes flow, anisotropic diffusion–reaction systems, elasticity problems, and moving interfaces.Source: §5 · p. 25
  • Future work will address theoretical analysis, high-dimensional and nonlinear extensions, and adaptive sampling or multilevel techniques.The stated aim is to enhance scalability and efficiency.Source: §5 · p. 25
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