A hybrid physics-informed neural network framework for solving high-dimensional fractional partial differential equations

Introduction

Fractional differential equations have emerged as fundamental tools for modeling complex physical systems where non-local interactions and memory effects play a crucial role. Applications span anomalous diffusion in heterogeneous media, viscoelasticity in materials science, signal processing, financial mathematics, and biological systems. The fractional Laplacian, defined as a pseudo-differential operator with symbol |ξ|α|, captures long-range interactions that classical integer-order operators cannot represent. Despite their theoretical elegance, the numerical solution of fractional PDEs presents formidable computational challenges. The non-local nature of fractional operators requires information from the entire domain, leading to dense linear systems with O(N 2 ) storage and O(N 3 ) computational complexity in one dimension, with exponential growth in higher dimensions. Traditional methods such as finite differences, finite elements, and spectral methods face severe limitations when applied to high-dimensional problems due to the curse of dimensionality. Physics-Informed Neural Networks (PINNs), introduced by Raissi et al., have emerged as a promising meshfree alternative for solving differential equations. By embedding governing physical laws directly into the neural network training process, PINNs can overcome the limitations of traditional numerical methods. However, the application of PINNs to fractional PDEs introduces additional challenges: the non-local nature of fractional operators complicates the computation of residuals, and automatic differentiation (AD) cannot be directly applied to fractional derivatives as the classical chain rule does not hold. Recent advances have extended PINNs to fractional problems. Pang et al. introduced fPINNs using Monte Carlo approximation of fractional operators. Guo et al. extended this to tempered fractional PDEs with the MC-tfPINN method. However, Monte Carlo-based approaches suffer from significant variance and errors, affecting convergence and exhibiting sensitivity to hyperparameters. Hu et al. demonstrated that replacing one-dimensional Monte Carlo integration with Gaussian quadrature can significantly improve accuracy and convergence speed in very high dimensions, scaling up to 100,000 dimensions. This paper presents a comprehensive hybrid PINN framework that addresses these challenges through five key contributions:

A fully-connected network with 4 hidden layers and 64 neurons per layer using Tanh activation, optimized for high-dimensional fractional PDEs.

A principled weighting strategy based on Neural Tangent Kernel theory that automatically balances multiple loss components, eliminating manual tuning.

Integration of Gaussian quadrature for fractional Laplacian approximation, providing deterministic error bounds and reduced variance compared to Monte Carlo sampling.

Combination of first-order global exploration with second-order local refinement for efficient training.

Systematic evaluation across five benchmark problems from 2D to 20D, establishing performance metrics and demonstrating scalability.

Methodology

Problem formulation

We consider high-dimensional fractional partial differential equations of the form:

subject to boundary conditions 𝐵[𝑢 ( 𝑥, 𝑡 ) ] = 𝑔 ( 𝑥, 𝑡 ) on ∂Ω and initial conditions u(x,0)=u0(x), where Ftα is a fractional time-derivative operator of order αα, and NN is a differential operator potentially in-cluding fractional spatial derivatives such as the fractional Laplacian (-Δ)α/2. The fractional Laplacian is defined via the Riesz potential:

Neural network architecture

The proposed PINN employs a fully-connected feedforward neural network with:

Input Layer: Accepts spatial coordinates x∈R d and time t, normalized to [-1,1] Hidden Layers: 4 hidden layers with 64 neurons each, determined through systematic hyperparameter optimization Activation Function: Hyperbolic tangent (Tanh), selected for smoothness and compatibility with fractional derivative requirements Output Layer: Single neuron producing the solution approximation u θ (x,t) For problems with homogeneous Dirichlet boundary conditions, hard constraint enforcement is implemented using a ReLU-based multiplier: uθ(x)=ReLU(1-∥x∥2)⋅uˉθ(x)uθ (x)=ReLU(1-∥x∥2)⋅uˉθ (x) where uˉθuˉθ is the raw network output.

Physics-informed loss function

The composite loss function consists of three components: 𝐿 total ( 𝜃 ) = 𝜆 PDE + 𝜆 BC + 𝜆 LIC

NTK-adaptive loss weighting

The loss weights are dynamically adjusted based on Neural Tangent Kernel theory to balance gradient contributions. The NTK describes the evolution of the network output during training and provides theoretical guarantees for convergence under appropriate conditions. For fractional PDEs, the NTK must account for the differential operators applied to the network output, making it distinct from standard neural network NTK.

Fractional operator approximation

Automatic Differentiation. Integer-order derivatives are computed exactly through the computational graph using PyTorch's autograd engine, achieving machine-precision accuracy with RMSE of 1.34 × 10 - 6 . Fractional Laplacian. The fractional Laplacian is approximated using a geometry-adaptive decomposition:

where r 0 (x) is a spatially varying radius based on the distance to the boundary. The singular near-field integral is evaluated using Gauss-Jacobi quadrature, the regular far-field integral uses Gauss quadrature, and angular variables are sampled via Monte Carlo. This approach provides deterministic error bounds and reduced variance compared to pure Monte Carlo sampling. Temporal Fractional Derivatives. Caputo fractional derivatives are approximated using the L1-scheme with a memory-efficient implementation.

Hybrid optimization strategy

Precise local refinement using an approximated Hessian

The transition between optimizers is seamless: parameters from Adam serve as initialization for L-BFGS. This hybrid approach combines the complementary benefits of both methods -rapid initial convergence from Adam and precise refinement from L-BFGS.

Numerical results

Benchmark problems

The framework was validated using five benchmark problems spanning different fractional operator types and dimensions:

Benchmark

Error metrics

Relative L 2 Error: E_L 1.56 × 10 -3 2.12 × 10 -3 538.2 min The proposed framework demonstrates superior performance compared to MC-fPINN, achieving a 1.2-1.4× improvement through quadrature-based enhancements. The framework also handles higher dimensions than existing methods, with MC-fPINN typically limited to 10D while the proposed framework successfully handles up to 20D. The framework exhibits near-linear scaling: O(d^1.2) in training time and O(d^1.1) in memory usage, confirming effective mitigation of the curse of dimensionality. The hybrid optimization strategy achieves a total loss reduction of 4.63 × 10 3 ×, with the L-BFGS phase providing up to 15.8× faster convergence than Adam alone near the optimum.

Results summary

Comparison with existing methods

Scaling analysis

Hybrid optimization performance

Discussion

Key findings

Architectural Effectiveness. The optimal architecture -4 hidden layers, 64 neurons, Tanh activation -successfully scales from 2D to 20D while maintaining stable training and accurate predictions.

NTK-Adaptive Weighting. The converged weight ratio of 1.0: 3.1: 2.1 (PDE: BC: IC) confirms that boundary conditions require the strongest enforcement, reflecting the non-local nature of fractional operators.

Quadrature Enhancement. Replacing Monte Carlo sampling with Gaussian quadrature for the radial integral provides deterministic error bounds and reduces variance, yielding a 1.2-1.4× improvement over MC-fPINN.

Optimization Strategy. The hybrid Adam-L-BFGS approach effectively combines global exploration with local refinement, achieving a 4.63 × 10 3 × total loss reduction.

Comparison with literature

The proposed framework aligns with recent advances in fractional PINNs.

Limitations

1. Dimensional Limit: Validated only up to 20D; may not extend to d > 30-50 without architectural modifications.

2. Fractional Operator Types: Limited to standard and tempered fractional operators; variable-order operators are not addressed.

3.

Training Time: 9 hours for 20D problems may be challenging for very complex applications.

4.

Hyperparameter Sensitivity: Performance depends on optimal configuration requiring systematic tuning.

Conclusion

This paper has presented a hybrid Physics-Informed Neural Network framework for solving high-dimensional fractional partial differential equations. The framework integrates automatic differentiation for exact integer-order derivatives with quadrature-enhanced numerical approximation for non-local fractional operators. A Neural Tangent Kernel-based adaptive loss weighting strategy balances competing physical constraints, while a hybrid Adam-L-BFGS optimization strategy ensures efficient training.

Validation across five benchmark problems spanning two to twenty dimensions demonstrates relative L 2 errors from 3.12 × 10 -4 to 5.67 × 10 -3 , with near-linear scaling O(d^1.2) in training time. The framework consistently outperforms existing MC-fPINN methods by 1.2-1.4× through quadrature-based enhancements. Key contributions:

1. An optimal neural network architecture for high-dimensional fractional PDEs.

2. NTK-adaptive loss weighting for principled constraint balancing.

3. Quadrature-enhanced fractional operator approximation.

4. Hybrid Adam-L-BFGS optimization.

Comprehensive validation establishing practical applicability up to 20 dimensions.

Future work will explore adaptive sampling strategies, extension to extremely high-dimensional problems using tensor networks, uncertainty quantification through Bayesian PINNs, and application to realworld engineering problems involving anomalous transport and nonlocal interactions.

References

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