Multi-Stage PINN Framework

• Multi-Stage PINN is a framework that partitions temporal, spatial, or spectral domains to deploy cascaded neural networks for solving PDEs with high precision.
• It uses specialized architectures like dual-network techniques and spectral analysis to overcome convergence issues and spectral bias in stiff and multiscale problems.
• Empirical results demonstrate error reductions up to four orders of magnitude, matching machine precision and ensuring robust enforcement of boundary conditions.

The Multi-Stage Physics-Informed Neural Network (MS-PINN) framework comprises a family of neural PDE-solving strategies in which either the temporal, spatial, or residual domains of a target system are partitioned and addressed by a cascade of separately trained or iteratively fit neural networks. These methods have emerged to overcome bottlenecks in convergence, stability, and spectral bias that restrict the accuracy of single-stage PINNs, especially for stiff, multiscale, nonlocal, and high-gradient problems. MS-PINN techniques encompass fractional diffusion modeling, dual-network boundary-interior specialization, spectral separation of physical modes, and iterative spectral-prior correction schemes, yielding accuracy improvements of two to four orders of magnitude across diverse equation classes, often matching machine precision.

1. Motivations and Conceptual Overview

The impetus for MS-PINN lies in the fundamental limitations of single-network PINNs—particularly their inability to resolve disparate frequency components, extended memory effects, or sharp boundary layers in one stage. In time-fractional subdiffusion, standard PINNs are forced to fit both low and high-frequency solution features over full domains, resulting in optimization difficulties and relative errors typically limited to $10^{-3}$–$10^{-4}$ (Xue et al., 28 May 2025). MS-PINN schemes partition the domain (temporally, spatially, or spectrally) so that each subnetwork is responsible for a smaller, more regular subset of the problem. This decomposition reduces the effective operator stiffness, sharpens gradient flow during training, and facilitates local error correction, often in a sequential error-cascade.

For PDEs with sharp gradients and multiscale features, dual-subnetwork architectures (e.g., domain and boundary networks) further enable focused specialization, soft decoupling via distance-weighted priors, and improved boundary enforcement (Abbas et al., 28 Nov 2025). Advanced variants use spectral analysis of the residual to inform network initialization and feature selection, explicitly targeting the modes responsible for persistent error or slow convergence (Li et al., 25 Aug 2025, Qian et al., 1 Jan 2026, Wang et al., 2024).

2. Mathematical Formalism and Stagewise Algorithms

A canonical MS-PINN protocol divides the solution interval or domain into $K$ stages, each employing a subnetwork $u_k(x,t;\theta_k)$ to solve the PDE over its local subinterval and interface points. For fractional subdiffusion systems:

$$\frac{\partial^\alpha u}{\partial t^\alpha}(x,t) = \mathcal{L}_x[u](x,t) + f(x,t),\quad 0<\alpha<1$$

with Caputo time-derivatives and appropriate boundary/initial conditions, the domain $[0,T]$ is split so that each subnetwork only accesses a localized time-history. To guarantee global $C^0$-continuity, loss function components penalize stage boundary mismatches:

$$L_k(\theta_k) = \frac{1}{N_f}\sum_{i=1}^{N_f}|\mathcal{N}[u_k](x_i,t_i)|^2 + \frac{\lambda}{N_b}\sum_{j=1}^{N_b}|u_k(x_j,t_{k-1};\theta_k) - u_{k-1}(x_j,t_{k-1};\theta_{k-1})|^2$$

Total loss is additive across stages. Both sequential and joint (parallel) training regimens are supported, with interface regularization ensuring solution continuity (Xue et al., 28 May 2025).

In spectral-prior-guided MS-PINNs, the residual of each stage is analyzed by discrete Fourier transform (DFT), extracting dominant frequencies and amplitudes to initialize embedding layers of the next stage. Subsequent networks fit the normalized residual recursively:

$u^{(n)}(x) = u^{(n-1)}(x) + \epsilon_{n-1}u_n(x)$

where $\epsilon_{n-1}$ is the RMS of the previous residual (Li et al., 25 Aug 2025). Alternative implementations sample random Fourier features according to the residual’s power spectral density, dynamically guiding feature selection in each stage.

For nonlinear compressible flow in infinite domains, a coordinate transformation compacts the domain, and multi-stage networks iteratively correct residuals, introducing spectral scaling and error weighting to efficiently drive errors to machine precision (Qian et al., 1 Jan 2026).

3. Specialized Architectures and Spectral Strategies

MS-PINN methods have been extended to address multi-scale and boundary-layer phenomena via domain decomposition and dual-network specializations. The Multi-Phase Dual-PINN decomposes the solution $u(x)$ into a domain network $u_D(x)$ and a boundary network $u_B(x)$:

$u(x) \approx u_D(x) + u_B(x)$

Loss terms combine unified physics residual, augmented Lagrangian boundary enforcement, and soft specialization through distance-weighted priors. The cosine-annealed role weights control specialization during distinct training phases:

$$\gamma(t) = \gamma_{min} + \frac12(\gamma_{max}-\gamma_{min})(1+\cos(\pi t/T))$$

This structure yields substantial reductions in mean absolute error (MAE), relative $L^2$ error, and boundary error compared to monolithic PINNs, with empirically observed 2–9$\times$ speedups in convergence (Abbas et al., 28 Nov 2025).

Spectral decomposition in multi-scale PINNs separates large-scale modes (treated by DNS, PINN, or coarse solver) and small-scale modes (learned exclusively by spectral PINN in frequency space). Residuals are propagated only from the large-scale solution, assuming negligible back-reaction. Pseudospectral techniques compute nonlinear terms, with network outputs matched directly to the slaved fine-scale spectrum (Wang et al., 2024).

Spectrum-informed multistage PINNs (SI-MSPINNs) and spectrum-weighted random Fourier feature MSPINNs (RFF-MSPINNs) further combat spectral bias and high-frequency error, utilizing DFT-based dominant mode extraction and PSD-weighted random frequency sampling (Li et al., 25 Aug 2025).

4. High-Precision and Multiprecision Aspects

A specific challenge of fractional PDEs lies in catastrophic cancellation and round-off when computing memory-kernel weights for the Caputo derivative and related operators. MS-PINN frameworks resolve these issues via multiprecision arithmetic (128- or 256-bit floats):

• Fractional derivative weights and $\Gamma(1-\alpha)$ terms are computed in high-precision libraries or hardware.
• Network weights remain in standard 64-bit precision during initial phases, with conversion to higher precision for final fine-tuning.
• Empirically, this multiprecision approach reduces solution error floor from $10^{-3}$–$10^{-4}$ to $10^{-7}$–$10^{-8}$ (Xue et al., 28 May 2025).

5. Numerical Results and Empirical Validation

MS-PINN demonstrates quantitative advantages across problem classes:

Equation / Mesh	PINN Error (Single)	MS-PINN Error (Stage 2)	Error Reduction
Fractional Exponential, Uniform (α=0.5)	$1.51\times 10^{-3}$	$3.92\times 10^{-6}$	$>10^3\times$
Fractional Polynomial, Graded (α=0.9)	$2.67\times 10^{-4}$	$8.74\times 10^{-8}$	$>10^3\times$
Laplace 2D (MAE) (Abbas et al., 28 Nov 2025)	$1.78\times 10^{-2}$	$9.7\times 10^{-3}$	$46\%$
Poisson 2D (MAE) (Abbas et al., 28 Nov 2025)	$4.56\times 10^{-3}$	$1.02\times 10^{-3}$	$78\%$
Burgers (L₂ loss) (Li et al., 25 Aug 2025)	$1.22\times 10^{-3}$	$7.34\times 10^{-7}$	$>10^3\times$
Helmholtz (L₂ error, real) (Li et al., 25 Aug 2025)	$9.2\times 10^{-3}$	$2.9\times 10^{-6}$	$>10^3\times$

MS-PINN techniques achieve error reductions of up to four orders of magnitude versus single-stage PINNs, robust boundary enforcement, and improved near-boundary accuracy. Spectral-prior-guided schemes demonstrate special efficacy for high-frequency, high-contrast scenarios. Compressible flow benchmarks using compactified infinite-domain MS-PINN strategies validate elimination of truncation artifacts and explicit quantification of linearization error, with machine precision residuals for both Laplace and nonlinear compressible equations (Qian et al., 1 Jan 2026).

6. Limitations, Extensions, and Practical Considerations

MS-PINN frameworks assume weak feedback between network stages in decoupled spectral approaches. Situations with strong backward coupling may require joint stage optimization or memory-term augmentation (e.g., via Mori-Zwanzig formalism) (Wang et al., 2024). Dual-network architectures depend on effective annealing schedules and sampling strategies to avoid role collapse. In fractional and stiff systems, mesh grading improves initial-stage accuracy, but may induce round-off sensitivity, especially at extreme parameter regimes (Xue et al., 28 May 2025). High-dimensional small-scale modeling is managed by aggressive mode truncation or decomposed network hierarchies.

All MS-PINN methodologies are mesh-free, compatible with arbitrary geometries and non-uniform meshes, and have demonstrated robustness in the presence of moderate data noise. Public code for spectrum-prior-guided MS-PINNs is available, supporting reproducibility and adaptation to new high-accuracy PDE settings (Li et al., 25 Aug 2025).

7. Impact, Outlook, and Theoretical Implications

Multi-Stage PINNs constitute a general paradigm shift for neural PDE solvers, offering a modular hybrid of error-cascading, physical specialization, and adaptive feature construction. They are particularly well suited to fractional operators, nonlinear flow equations in unbounded domains, multiscale turbulence, and strong boundary-gradient regimes. Typical sample efficiency, convergence rate, and final accuracy are substantially improved over monolithic PINNs. Further research may address high-dimensional slaving, joint optimization under strong coupling, and dynamic curriculum schedules.

A plausible implication is that MS-PINN principles—spectral partition, multiphase role division, and iterative error correction—will underpin next-generation neural PDE solvers across scientific computing, with extensions to data-driven operator inference, multiphysics, and control. Trends suggest increasing adoption of multiprecision tensor computation and informed, residual-directed adaption in network initialization and training algorithms.