---
title: 'FDIFF-PINN: Fractional Differential Equation PINNs'
url: https://www.emergentmind.com/topics/fractional-differential-equation-physics-informed-neural-network-fdiff-pinn
type: topic
---

# FDIFF-PINN: Fractional Differential Equation PINNs

A Fractional Differential Equation Physics-Informed Neural Network (FDIFF-PINN) is a class of scientific machine learning frameworks that leverages deep neural network function approximators to solve direct, inverse, and parametric problems governed by fractional differential equations (FDEs). Distinct from integer-order PINNs, FDIFF-PINNs are constructed for equations involving Caputo, Riemann–Liouville, Grünwald–Letnikov, conformable, or Riesz derivatives—PDEs modeling memory, nonlocality, anomalous diffusion, or heavy-tailed processes—by hybridizing neural architectures with specialized quadrature or discretization for fractional operators. FDIFF-PINNs unify advances in automatic differentiation, fractional calculus, spectral and weak formulations, and stochastic representations to yield PDE-constrained learning algorithms applicable from deterministic subdiffusion to high-dimensional fractional Fokker–Planck–Lévy equations, parametric battery modeling, and random-field SFPDEs.

## 1. Mathematical Formulation and Governing Equations

Fractional PDEs generally have the form:
\[
\mathcal D_t^\beta u(x,t) + \mathcal L_x^{\alpha} u(x,t) + \mathcal M[u;\theta_p] = f(x,t),\quad (x,t)\in \Omega\times(0,T]
\]
where $\mathcal D_t^\beta$ is a time-fractional derivative (Caputo, Riemann–Liouville, conformable, G-L, etc.) of order $0<\beta<1$ or $0<\alpha<2$ for space, $\mathcal L_x^{\alpha}$ is a non-local spatial operator (Riesz/Caputo/tempered Laplacian), $\mathcal M$ encodes lower-order or nonlinear terms, and $f(x,t)$ is the source (potentially black-box or random). Boundary and initial conditions are problem-specific; for Dirichlet/zero-flux, hard/soft constraints or network reparameterization are used to satisfy them [1811.08967][2303.10913][2406.11708][2503.22386].

Specific examples include:
- **Time-fractional diffusion (Caputo form):**
  \[
  \frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\partial_s u(x,s)\,ds = \nabla\cdot(D(u)\nabla u(x,t)) + f(x,t)
  \]
- **Space-fractional Fokker-Planck–Lévy:** Involves the Riesz fractional Laplacian $(-\Delta)^{\alpha/2}$ for heavy-tailed processes [2406.11676][2406.11708].
- **Fractional battery models:** G-L derivatives describe non-local SOC and polarization voltage dynamics [2512.12285].
- **Parametric and stochastic SFPDEs:** Incorporate uncertainty via random fields, handled via orthogonal or bi-orthogonal expansions [2303.10913][2503.22386].

## 2. Discretization and Numerical Evaluation of Fractional Operators

Standard PINN automatic differentiation cannot natively handle fractional derivatives due to their functional nonlocality. FDIFF-PINNs circumvent this as follows:

- **Caputo/Grünwald–Letnikov (G-L) Time Derivatives:**
  \[
  D_t^\alpha u(t_n) \approx (\Delta t)^{-\alpha} \sum_{m=0}^n w_m^{(\alpha)} u(t_{n-m}),\quad w_m^{(\alpha)} = (-1)^m \frac{\Gamma(\alpha+1)}{\Gamma(m+1)\Gamma(\alpha-m+1)}
  \]
  Embedded either in the computation graph [2510.25925][2512.12285] or via finite-difference quadrature (L1/L2 schemes) [2407.09496][1811.08967].

- **Space-Fractional Laplacian (Riesz/Tempered):**
  - **Directional Grünwald–Letnikov:** Discretizes via dense convolutional stencils along angular directions (auxiliary grid) [1811.08967].
  - **Monte Carlo (MC-fPINN/MC-tfPINN):** High-dimensional spatial integrals are split into 1D radial (handled via Gauss–Jacobi/Laguerre quadrature) and $(d-1)$-dimensional spherical MC sampling, enabling up to $10^5$ dimensions [2406.11708].
  - **Score-based fractionalization:** Integrate-by-parts formulations introduce a learnable "fractional score function" to locally reparameterize the nonlocal operator, then solve an equivalent second-order PDE [2406.11676].

- **Conformable derivatives:** These admit a chain-rule-friendly form $T_\alpha[f](t) = t^{1-\alpha} f'(t)$, fully compatible with AD [2108.07490].

- **Spectral methods:** Global Legendre (or Chebyshev) expansions in space (and in time for multi-term FDEs), with NN learning modal coefficients, yielding mesh-free, fast convergence for smooth solutions [2503.22386].

## 3. Neural Network Architectures and Loss Construction

Most FDIFF-PINN approaches leverage multi-layer perceptrons (MLPs) or recurrent networks (for sequential data) as universal surrogates for $u(x,t)$, auxiliary coefficients (e.g., $D(u)$, $a(x)$), or bi-orthogonal stochastic modes. Variants include:

- **Hard Constraint Networks:** Multiply base NN by spatial masks (e.g., $\rho(x) = \max(0,1-\|x\|^2)$) and/or $t$ to enforce $u|_{\partial \Omega}=0$ and $u(x,0)=u_0(x)$ by construction [2406.11708][1811.08967].
- **Multi-output Architectures:** Separate networks for solution, coefficients, fractional order, and uncertainty modes [2303.10913][2407.09496].
- **Spectral Coefficient Nets:** DNN learns time and parameter dependence of spectral expansion coefficients [2503.22386].

The loss function generically aggregates:
\[
\mathcal L_\mathrm{total} = \lambda_\text{resid} \mathcal L_\text{PDE} + \lambda_\text{data} \mathcal L_\text{data} + \lambda_\text{IC/BC} \mathcal L_\text{IC/BC} + \lambda_\text{prior} \mathcal L_\text{prior}
\]
- **$\mathcal L_\text{PDE}$:** Mean-squared residuals of the FDE at collocation points, fractional discretization included [2406.11708][1811.08967].
- **$\mathcal L_\text{data}$:** MSE on available observational data (SOC in batteries, concentration, etc.) [2512.12285][2407.09496].
- **$\mathcal L_\text{IC/BC}$:** Strong or weak imposition of initial/boundary data.
- **$\mathcal L_\text{prior}$:** Parameter or coefficient priors for parameter identification or regularization [2304.00909].

Inverse problems treat physical parameters (fractional order $\alpha$, conductivity $k$, coefficient fields) as trainable variables, backpropagating through all loss terms [2510.25925][2512.12285][2407.09496][2304.00909].

## 4. Algorithmic Workflow and Training Strategies

A generic workflow is as follows:
1. **Sampling:** Uniform/random selection of collocation points for the PDE (space–time), and additional data or sensor points for inverse tasks.
2. **Forward Pass:** For each batch, evaluate NN predictions at required points, compute all AD and discretized derivatives.
3. **Residual and Loss Evaluation:** Compute the physics-informed residuals, data terms, and any regularization according to the selected loss function.
4. **Backward Pass:** Gradients w.r.t. all NN weights and trainable PDE parameters (fractional order, diffusion, etc.) computed by AD.
5. **Optimizer Steps:** Usually Adam for initial epochs, transitioning to second-order (L-BFGS-B) for fine-tuning. Learning rates and schedules are often problem-dependent [2510.25925][1811.08967][2303.10913].
6. **Postprocessing:** For Laplace-based or score-based FDIFF-PINNs, post-training inversion or quasi-static projection may be needed ([Laplace inversion via Stehfest][2304.00909], spectral coefficient evaluation [2503.22386], or score function root-finding [2406.11676]).

A condensed pseudocode for high-dimensional MC/Quadrature-based FDIFF-PINNs appears below (from [2406.11708]):
```python
# Precompute quadrature and MC sampling setup

for epoch in range(N_epochs):
    # Sample collocation, initial/boundary, (and data) points
    for each collocation point:
        # Evaluate u_pred, compute time/space-fractional terms via quadrature and/or MC
        # Form physics residuals, IC/BC residuals
    Compute losses L_total
    Backpropagate via AD
    Update θ, and trainable physical parameters
```

## 5. Representative Applications and Numerical Benchmarks

FDIFF-PINNs have been rigorously validated across benchmark and real-world problems:

### Benchmark Summaries

| Paper                  | Problem Class                 | Dimension(s) | Operator Type | Method/Discretization           | Rel. Error(s)          |
|------------------------|------------------------------|--------------|--------------|---------------------|------------------------|
| [1811.08967]           | Forward/inverse ADE           | 1D/2D/3D     | Space/Time    | GL, L1                | $10^{-3}$–$10^{-4}$   |
| [2406.11708]           | High-dim Poisson/diffusion    | up to $d=10^5$| Riesz/tempered| MC, Gauss–Jacobi      | $10^{-3}$–$10^{-2}$   |
| [2503.22386]           | Parametric FDEs               | 1D           | Time          | Legendre-Galerkin + DNN      | $10^{-3}$–$10^{-4}$   |
| [2512.12285]           | Lithium-ion battery           | 1D (time)    | G-L time      | G-L, MLP/RNN/LSTM         | MSE reduced 30–80% wrt baseline |
| [2407.09496]           | Inverse anomalous diffusion/viscoelasticity | 2D/1D  | Caputo time    | L1-scheme, Swish NN      | rel. err. <10% under 25% noise |

Further, specialized strategies (Laplace-fPINN, score-PINN, spectral coefficient learning, bi-orthogonal expansions) are documented for fractional heat conduction [2510.25925], subdiffusion [2304.00909], high-dimensional Fokker–Planck–Lévy [2406.11676], and stochastic SFPDEs [2303.10913].

### Notable Outcomes

- **Generalization and Physical Consistency:** FDIFF-PINNs with physics-informed loss yield models robust to severe measurement noise, generalize across parametric ranges, and recover parameters/fractional orders with precision approaching 0.1% [2510.25925][2407.09496][2512.12285].
- **Curse of Dimensionality:** MC-fPINN and score-fPINN frameworks mitigate exponential scaling of curse of dimensionality, enabling tractable training and inference in $d=10^3$–$10^5$ [2406.11708][2406.11676].
- **Stochastic Fractional PDEs:** BO-fPINN effectively manages mode-crossing and uncertainty quantification in SFPDEs, outperforming plain bi-orthogonal and gPC methods in high dimensions [2303.10913].

## 6. Theoretical Guarantees, Limitations, and Extensions

- **Convergence:** Whenever spectral discretizations are used, convergence rates are theoretically quantified, decomposing error into spectral truncation, DNN approximation, and generalization error terms, all vanishing in appropriate limits [2503.22386].
- **Limitations:** FDIFF-PINNs require careful balancing of hybrid loss weights, memory lengths/$\alpha$ for G-L discretization, and may exhibit reduced stability for ill-posed or highly dynamic regimes (e.g., complex battery loading in [2512.12285]). Spatial operators relying on finite differences are not fully mesh-free and may underperform for non-smooth solutions [2303.10913].
- **Computational burden:** Nonlocal memory in time (Caputo/G-L) and space (Riesz, tempered) can become a bottleneck for large histories or high accuracy, but MC/quadrature hybridization yields practical algorithms [2406.11708].
- **Future directions:** Promising avenues include domain decomposition (XPINN), operator learning (DeepONet, Fourier-NO), adaptive loss balancing, transfer learning for fractional order, probabilistic (Bayesian) extensions, and efficient implementation of meshless fractional operators [2406.11708][2406.11676][2303.10913][2512.12285].

## 7. Variants, Modalities, and Methodological Synthesis

FDIFF-PINNs are not a monolithic architecture but a broad family unified by physics-based loss and explicit numerical accommodation for fractional operators. Variants include:

- **Laplace-fPINN:** Transforms Caputo-type time-fractional PDEs to elliptic problems in $(x,s)$-Laplace domain, enabling use of classic PINNs and avoiding time-history convolutions [2304.00909].
- **Score-fPINN:** Transforms fractional spatial operators to local PDEs for the log-likelihood by introducing a fractional score (via integration by parts), facilitating mesh-free high-dimensional solution [2406.11676].
- **Spectral PINN:** Employs trial solutions as NN-generated linear combinations of global basis functions (e.g., Legendre), eliminating spatial automatic differentiation and delivering provably optimal spectral convergence in parametric settings [2503.22386].
- **Bi-orthogonal fPINN:** Decomposes stochastic FDE solutions into modal expansions, with all bi-orthogonal constraints imposed weakly via the loss, allowing for both forward/inverse stochastic solution and transfer learning [2303.10913].
- **Conformable PINN:** For equations with conformable derivatives, exploits the AD-friendly structure of the operator to preserve full meshless convenience [2108.07490].

Each branch can be adapted for inverse parameter identification, data assimilation, discovery of anomalous exponents, or uncertainty quantification, as dictated by problem requirements [2510.25925][2407.09496].

---

**References:**
- [1811.08967] fPINNs: Fractional Physics-Informed Neural Networks
- [2108.07490] Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative
- [2303.10913] Bi-orthogonal fPINN: A physics-informed neural network method for solving time-dependent stochastic fractional PDEs
- [2304.00909] Laplace-fPINNs: Laplace-based fractional physics-informed neural networks for solving forward and inverse problems of subdiffusion
- [2406.11676] Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations
- [2406.11708] Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks
- [2407.09496] Physics-Informed Neural Network based inverse framework for time-fractional differential equations for rheology
- [2503.22386] Spectral coefficient learning physics informed neural network for time-dependent fractional parametric differential problems
- [2510.25925] Equation Discovery, Parametric Simulation, and Optimization Using the Physics-Informed Neural Network (PINN) Method for the Heat Conduction Problem
- [2512.12285] Fractional Differential Equation Physics-Informed Neural Network and Its Application in Battery State Estimation

Source: https://www.emergentmind.com/topics/fractional-differential-equation-physics-informed-neural-network-fdiff-pinn