Source-linked AI summary
The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?
Stefano Markidis
TL;DR
The paper asks whether PINNs can solve Poisson-equation linear systems as alternatives to traditional methods. It evaluates PINN configurations and transfer learning, then integrates PINNs into traditional solvers; the resulting hybrid solvers are on par with PETSc conjugate-gradient solvers in performance and accuracy. Direct PINN use remains limited by accuracy and computational costs.
Problem
The paper evaluates whether PINNs can replace traditional solvers for linear systems arising from the Poisson equation, an important scientific-computing problem.
Method
The study characterizes PINN solver behavior across network configurations and transfer learning, then embeds a PINN linear solver within a multigrid V-cycle with traditional refinement.
Results
Hybrid solvers integrating PINNs with traditional methods are on par with PETSc conjugate-gradient solvers in performance and accuracy.
Takeaways & Limitations
Hybrid strategies combining deep-learning techniques with traditional linear solvers are among the most promising approaches for developing a new class of linear solvers.
Takeaways & Limitations
Direct PINN solvers remain limited by accuracy and long training periods, while L-BFGS-B often dominates hybrid-solver execution time.
Abstract
from arXiv · showhide
Physics-Informed Neural Networks (PINN) are neural networks encoding the problem governing equations, such as Partial Differential Equations (PDE), as a part of the neural network. PINNs have emerged as a new essential tool to solve various challenging problems, including computing linear systems arising from PDEs, a task for which several traditional methods exist. In this work, we focus first on evaluating the potential of PINNs as linear solvers in the case of the Poisson equation, an omnipresent equation in scientific computing. We characterize PINN linear solvers in terms of accuracy and performance under different network configurations (depth, activation functions, input data set distribution). We highlight the critical role of transfer learning. Our results show that low-frequency components of the solution converge quickly as an effect of the F-principle. In contrast, an accurate solution of the high frequencies requires an exceedingly long time. To address this limitation, we propose integrating PINNs into traditional linear solvers. We show that this integration leads to the development of new solvers whose performance is on par with other high-performance solvers, such as PETSc conjugate gradient linear solvers, in terms of performance and accuracy. Overall, while the accuracy and computational performance are still a limiting factor for the direct use of PINN linear solvers, hybrid strategies combining old traditional linear solver approaches with new emerging deep-learning techniques are among the most promising methods for developing a new class of linear solvers.
1 INTRODUCTION
The introduction evaluates PINNs as solvers for Poisson-equation linear systems and motivates hybrid methods that combine PINNs with traditional solvers. It outlines PINN design factors, transfer learning, and the resulting accuracy and performance trade-offs.
- Motivation: The work evaluates PINNs as replacements for or complements to traditional methods for linear systems arising from the Poisson equation.The study targets an omnipresent PDE whose traditional solution can require non-negligible computational time.
- PINN formulation: PINNs train neural networks by minimizing governing-equation residuals and evaluate differential operators using automatic differentiation at collocation points.Basic PINN training is unsupervised and does not require labeled simulation or experimental data.
- PINN formulation: After training, PINN inference maps independent variables such as time and position to approximate governing-equation solutions without requiring a mesh.The trained network can predict on grids with different resolutions without retraining.
- Evaluation factors: The study varies network depth, activation functions, and input-data distribution, finding that activation choice strongly affects accuracy and convergence according to source-term smoothness.Fully connected networks deeper than three layers show similar performance during the first thousand training epochs.
- Transfer learning: Transfer learning initializes a PINN using training results from the same PDE with a different source term and considerably speeds network training.This setting matches applications that solve the same PDE with different source terms at successive time steps.
- Hybrid solvers: Direct PINN replacement remains uncompetitive with traditional solvers, whereas hybrid integration with multigrid and iterative methods produces solvers on par with PETSc conjugate-gradient solvers in performance and accuracy.The proposed hybrid approach combines a deep-learning solver on coarse grids with traditional refinement methods.
2 THE NEW: PHYSICS-INFORMED LINEAR SOLVERS
PINNs solve differential equations by combining a trainable surrogate network with an untrained residual network that encodes governing equations and supplies the optimization loss. For Poisson problems, they resolve low-frequency solution features first, while high-frequency accuracy remains slow and motivates hybridization with traditional solvers.
- Scientific-computing context: PINNs are evaluated for the Poisson equation, a widely solved elliptic PDE whose scientific applications include electrostatics, potential flow, pressure solvers, and CFD.Traditional elliptic-problem approaches include Krylov solvers and FFT methods.
- PINN formulation: PINNs approximate differential-equation solutions by training a surrogate network to minimize residuals produced by a network encoding the governing equations.The surrogate maps collocation points to approximate solutions, while the residual network supplies the loss during training.
- PINN formulation: The residual network uses automatic differentiation to compute differential operators, avoiding finite-difference derivative approximations.For Poisson problems, the Laplacian is formed from successive derivatives in the x and y directions.
- PINN formulation: During inference, only the surrogate network is used to calculate the problem solution.The residual network participates only in training by calculating the residual.
- Poisson example: The F-principle causes low-frequency, large-scale solution features to emerge before high-frequency, small-scale features during PINN training.In the example, diagonal low-frequency structure appears before localized high-frequency peaks are resolved.
- Hybrid solvers: Hybrid PINN-traditional solver designs target performance comparable to state-of-the-art linear solvers such as petsc4py.The paper proposes this integration because direct PINN accuracy and computational performance remain limiting factors.
3 CHARACTERIZING PINNS AS LINEAR SOLVERS
The paper characterizes fully connected PINN linear solvers across network size, training setup, source-term smoothness, and transfer learning. Depth and architecture choices strongly affect training, with four hidden layers offering the reported accuracy–cost trade-off while transfer learning accelerates reuse across source terms.
- Transfer learning: Transfer learning initializes PINN weights from a Poisson solve with a different source term, and the paper reports that it considerably speeds training.The study considers source terms with different smoothness and uses transfer learning to reuse previously obtained network weights.
- Network depth: One- and two-hidden-layer PINNs perform poorly, whereas networks with more than three layers achieve lower final training errors and improved learning.A single hidden layer with 640 units does not outperform deeper configurations, indicating that depth is more important than breadth in these tests.
- Network depth: Training error saturates beyond six hidden layers, limiting the benefit of additional depth.This saturation was reported for the two test cases considered.
- Network depth: Four or more hidden layers have comparable performance during the first 500 Adam and L-BFGS-B epochs.Because computational cost increases with layer count, four hidden layers provide the reported accuracy–performance trade-off.
encoding the low-frequencies components (fewer points are needed to represent low-frequency signals)
PINN training error depends strongly on activation function and data configuration. Adaptive activations improve training, while data distribution has limited impact for large datasets and retraining with larger data can worsen error.
- The activation function is the most impactful parameter for achieving low training error.
- Local adaptive activation functions reduce training error and are introduced by optimizing a separate scalable parameter for each layer.The paper connects these functions to mitigating the vanishing-gradient problem.
- Smooth source terms perform best with locally adaptive tanh, whereas non-smooth source terms perform best with locally adaptive sigmoid.The reported best configurations are LAAF5-tanh for smooth sources and LAAF10-sigmoid for non-smooth sources.
- For large datasets, Sobol and pseudo-random collocation distributions perform slightly better than uniform, while pseudo-random performs better for small datasets.The study compares uniform, pseudo-random, and Sobol distributions across three dataset sizes.
- Retraining a PINN from a small dataset on a large dataset does not decrease error and produces the highest training error.The tested restart procedure first used 1,200 domain and 200 boundary points, then 128×128 domain and 4,000 boundary points.
4 THE IMPORTANCE OF TRANSFER LEARNING
Transfer learning substantially accelerates PINN training by initializing a network with weights learned for the same PDE and a different source term. Its benefit depends on choosing a suitable pretrained problem, especially one containing high-frequency source components.
- Transfer learning initializes a PINN with weights and biases learned from solving the same PDE with a different source term.The first trained network transfers learned information about encoding to the second PINN.
- Using transfer learning enables L-BFGS-B after very few Adam epochs; empirically, 10 Adam epochs avoid convergence to a local minimum.Without transfer learning, training uses 2,000 Adam epochs followed by 5,000 L-BFGS-B epochs; with it, the reported schedule uses 10 Adam epochs followed by 6,955 L-BFGS-B epochs.
- Transfer learning gives two orders of improvement in training error in less than 1,000 epochs.The comparison uses Xavier initialization without transfer learning versus initialization from a previously trained network.
- Transfer learning rapidly decreases training error for smooth-source and non-smooth-source Poisson test cases.Figure 6 compares training with and without transfer learning for both source-term types.
- Pretrained networks containing high-frequency source components are generally suitable for transfer learning, whereas single low-frequency components provide no benefit over training without transfer learning.The paper identifies selecting an appropriate pretrained PINN as the major challenge.
5 THE OLD AND THE NEW: INTEGRATING PINNS INTO TRADITIONAL LINEAR SOLVERS
Direct PINN solvers converge quickly on low-frequency solution components but are slow for high-frequency accuracy. The paper therefore combines a PINN coarse-grid solve with Gauss-Seidel refinement and multigrid interpolation, producing competitive accuracy and performance.
- PINNs converge quickly to low-frequency components, while high-frequency convergence is slow and requires increasing training iterations.This frequency-dependent behavior motivates combining PINNs with traditional iterative solvers.
- The hybrid solver uses a PINN on a coarse grid, then Gauss-Seidel refinement and a multigrid V-cycle with interpolation to finer grids.The PINN replaces multigrid restriction or injection for the coarse-grid solution.
- L-BFGS-B typically dominates hybrid-solver execution time, limiting the computational advantage of the approach.
- The hybrid multigrid solver reaches maximum error of order 1E-4, below the approximately 1E-3 error reported for an extensively trained basic PINN.The error comparison is reported for the tested hybrid configurations with δ values of 1E-5 and 1E-4.
- For larger coarse grids such as 128×128, the hybrid multigrid solver is fastest, while Gauss-Seidel is faster than the DL solver on a 64×64 coarse grid.The comparison includes petsc4py conjugate-gradient results with different r_tol values.
- The hybrid solver is reported as competitive with state-of-the-art linear solvers, including the PETSc conjugate-gradient comparison.The authors state that none of the methods or codes were optimized or compared at identical conditions.
6 DISCUSSION AND CONCLUSION
The discussion finds that direct PINN use remains limited by accuracy, training time, and computational performance, while hybrid PINN–traditional approaches show strong promise for new linear solvers.
- Limitations: Direct PINN replacement produces limited accuracy and requires a long training period.These limitations motivate integrating deep-learning methods with traditional solvers rather than using PINNs alone.
- PINN optimization: PINN configuration choices and transfer learning can effectively optimize solver performance.The examined choices include network depth, activation functions, and input data-set distribution.
- Limitations: PINNs still cannot replace traditional HPC solvers such as PETSc CG in their current state.The paper identifies direct PINN accuracy and computational performance as limiting factors.
- Conclusion: The paper concludes that hybrid strategies combining PINNs with methods such as multigrid and Gauss-Seidel are the most promising route to new scientific solvers.This integration is proposed for next-generation solvers for linear systems arising from differential equations.
- Hybrid solvers: Hybrid solvers achieved performance on par with, and better performance for large coarse grids than, other state-of-the-art implementations such as PETSc.The paper presents these as initial performance results for hybrid approaches.
- Future performance: GPU use could allow hybrid solvers to outperform state-of-the-art HPC solvers, although the key L-BFGS-B optimizer lacks a SciPy GPU implementation.Tensor cores and TPUs are also identified as possible ways to boost deep-learning training performance.