TL;DR
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.
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.
This work embeds feature interaction modules derived from factorization machines (FMs) into physics-informed neural networks (PINNs) and neural operator learning, to enhance model expressiveness for solution manifolds of parameterized partial differential equations (PDEs). Motivated by the second-order Taylor expansion of multivariate functions to characterize variable couplings, we first propose FM-PINN. It explicitly captures spatio-temporal variable interactions and improves the approximation accuracy for smooth high-order PDEs. We further group spatial coordinates, time, physical parameters, and initial and boundary conditions into independent feature sets and model their cross-group interactions. Based on this strategy, we develop FM-Operator and FM-DeepONet, which are particularly effective for nonlinear conservation laws and problems with sharp gradients or discontinuities, while offering no consistent advantage on smooth operator learning benchmarks. Numerical tests demonstrate that the proposed mechanism delivers substantial accuracy gains on challenging shock-dominated equations, indicating a promising direction for physics-consistent modeling of parameterized PDEs with strong cross-field dependencies.
This work systematically compares two state-of-the-art frameworks-Physics-Informed Neural Networks (PINNs) and Optimizing a Discrete Loss (ODIL) across benchmark elliptic, hyperbolic, and parabolic problems, culminating in a challenging inverse source reconstruction task.
Persistent late-time variation can remain in physics-informed neural network (PINN) solutions after the governing transient has effectively decayed, while conventional error norms do not reveal whether this variation has a systematic temporal–frequency structure. This study develops an offline temporal–spectral diagnostic and postprocessing workflow for a one-dimensional advection–diffusion benchmark. A high-accuracy analytical reference and three fixed-resolution finite-difference baselines are used to assess a PINN whose architecture is selected by a fully supervised neural architecture search and whose parameters are trained with progressive temporal windowing. Candidate late-time intervals are selected without using the reference solution by applying the Bayesian information criterion (BIC) to a breakpoint model for the inter-reconstruction sensitivity; the selected field is subsequently reconstructed by retaining a prescribed fraction of its temporal spectral energy and is evaluated independently through reference-error and physics-consistency measures. For [tcut,tmax]=[1.8,5], the zero-frequency component contains 0.9999996 of the raw-field energy, so the q=0.95 reconstruction retains only the temporal mean. This projection reduces the final-time spatial error norm from 2.70×10−3 to 1.06×10−3, a factor of approximately 2.5, while changing the discrete governing-equation residual by less than 0.3% over the filtered window. Mean-removed tests for q=0.90,0.95,0.99 show that the discarded fluctuation is dominated by low-frequency approximation error rather than high-frequency noise. The result supports the proposed selection–validation workflow for this controlled benchmark but does not establish a universally transferable filter.
In this research, we provide a convergence of the Adomian decomposition based neural network method for time fractional reaction diffusion initial boundary value problems in 2D, where the fractional time derivative is considered in the Caputo sense. We propose an idea of a proper combination of Adomian decomposition method (ADM) and Physics-informed neural network (PINN), after semi-discretization of the time fractional derivative. PINN uses the ADM based partial sum inside the loss function before generating the appropriate ADM-PINN approximation. In addition, we produce the required sufficient conditions on the given data under which the usual Adomian decomposition method converges in a bounded domain. Furthermore, we analyze the error bound of the proposed method for the time fractional model and demonstrate the convergence. Several numerical examples are produced to show the effectiveness of the present ADM-PINN approach. It is observed that the proposed method is highly effective for several two-dimensional time fractional problems including the Schrodinger equation for convergent approximation of the computed solution.
This study introduces a dual-branch, spectrally-gated architecture (DBSG-PINN) that splits low- and high-frequency components into separate subnetworks joined by an adaptive gate, and uses it to run a partially controlled ablation of frequency decomposition and spectral routing.
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