PHD-SF establishes a separated-solving, hyper-reduction-like enrichment, and direct-inference acceleration path, reducing the dependence on large-scale high-fidelity data generation and its offline cost.
Abstract
Efficient solution of time-dependent parametric partial differential equations (PDEs) is central to computational science and engineering. Existing deep-learning-based accelerators span a physics-to-data spectrum, from physics-informed solvers with strong physical consistency but high computational cost to data-driven surrogates with efficient inference but strong dependence on high-fidelity datasets. These methods have largely evolved in isolation and are often connected only through explicit solution-field labels. We propose PHD-SF, a compact physics-to-data spectrum framework for accelerating time-dependent parametric PDEs. By combining an SD-TPD separation strategy with differential-operator learning, PHD-SF allows reusable model information, including spatial features, latent dynamical features, and spatial differential operators, to be generated, inherited, and enriched across three operating modes, PIDON, HIDON, and DIDON, within a unified DON-based architecture. This design enables label-free spectrum modeling without precomputed full-field solution labels or explicit label transfer among modes. PHD-SF further establishes a separated-solving, hyper-reduction-like enrichment, and direct-inference acceleration path, reducing the dependence on large-scale high-fidelity data generation and its offline cost. Results on four benchmark time-dependent PDEs show accurate cross-parameter solution and direct inference using only one or two representative parameter cases in the initial physics-informed stage. The total end-to-end cost is lower than that required to train a conventional PINN for a single parameter case, while supporting reusable cross-parameter inference. PHD-SF therefore provides an efficient solution path for many-query parametric PDEs.
Meta-SPINN works both as a direct predictor for unseen tasks and as a task-aware initializer for subsequent single-instance residual-guided refinement, providing reusable predictions together with an interpretable visualization of how solution geometry changes across a parameter family.
In this study, we introduce latent PDE mapping, a broadly applicable physics-informed learning technique designed to enable efficient geometric generalization with sparse training data. Latent PDE mapping pulls back geometry-specific PDE residuals and boundary conditions to a predefined latent geometry via the deformation gradient, thereby enabling the automated calculation of geometry-consistent shape gradients that are missing in conventional physics-informed machine learning formulations. We demonstrate the utility of latent PDE mapping in solving the anisotropic Aliev-Panfilov PDE of cardiac electrophysiology using both physics-informed neural networks and physics-informed deep operator networks. The Aliev-Panfilov PDE serves as a challenging exemplar: a nonlinear, time-dependent PDE benchmark with sharp gradients that are expensive to capture using traditional numerical solvers. To represent the limited data regime, we train the networks using just fifteen geometric samples drawn from parameterized distributions in two and three spatial dimensions. While modest improvements appear for geometries parameterized by affine and shear deformations, latent PDE mapping demonstrates significant benefits on select geometric families, achieving a factor ~4-6 reduction in mean relative L2 error. Furthermore, our results show that the computational cost of applying latent PDE mapping was modest during network training, and negligible at inference. Taken together, our study highlights how latent PDE mapping facilitates the creation of generalizable physics-informed machine learning models from limited sets of training geometries.
I. A. Adde, M. Maleckar, Gabriel Balaban· 0 citations
Extensive numerical experiments demonstrate that PINNs-MSFF achieves superior accuracy, stability, and convergence, effectively capturing complex fractional dynamics, sharp localized gradients, and dispersive phase transitions where standard PINNs often fail.
Harender Kumar· International Journal of Dyn...· 0 citations
This work proposes a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs that achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.
Pinki Khatun, M. Sajid, Abhinav Jha et al.· arXiv.org· 0 citations
The results demonstrate that PINN achieves more accurate and stable full-field vibration reconstructions than conventional PINNs, particularly under conditions involving high-frequency modes, and highlights the potential of hybrid data-physics neural frameworks as an efficient and reliable approach for solving complex PDE-governed dynamical systems.
Hai-Long Liu, S. Hedayatrasa, Yunpeng Zhu et al.· e-Journal of Nondestructive...· 0 citations
Accurate simulation of the long-time evolution of systems governed by partial differential equations (PDEs) is central to scientific computing. Among existing deep learning?based approaches for solving PDEs, neural operators typically rely on extensive trajectory data, whereas physics-informed meth?ods often exhibit limited stability during long-time extrapolation. For a well-posed autonomous PDE, long-time trajectories can be generated by repeated composition of a fixed-step evolution operator; hence, long-time extrapolation depends on controlling the approximation error of this operator and the propagation of that error under recursive composition. Accordingly, we propose a numerical-prior-guided, physics-constrained method trained without ground-truth trajectory supervision: a low-cost numerical prior reduces the difficulty of approximating the one?step evolution operator, while a weak-form PDE residual provides a computable proxy for the one-step error term in the error?propagation bound. We validate the method on five benchmark cases spanning four PDE classes and compare it with ten physics?informed learning methods under a unified protocol that excludes ground-truth trajectories from training and model selection. The results indicate that, in all five cases, the proposed method reduces long-time extrapolation error relative to the numerical prior and outperforms the best competing baseline in each case, thereby improving long-time simulation accuracy across different PDEs without ground-truth trajectory supervision. The source code developed for this paper will be made publicly available upon acceptance of the manuscript.
Maqun Zhang, Feng Gao, Wankun Chen et al.· 0 citations
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