Laplace-fPINNs for Subdiffusion Problems

• Laplace-fPINNs are a computational framework that transforms time-fractional diffusion equations into an algebraic Laplace domain, simplifying both forward and inverse subdiffusion problems.
• The methodology removes time-history dependence of Caputo derivatives, enabling standard neural network architectures and reducing computational costs.
• Laplace-fPINNs achieve high accuracy and scalability in multi-dimensional settings while facilitating parameter recovery through an advanced hierarchical loss system and numerical Laplace inversion.

Laplace-based fractional physics-informed neural networks (Laplace-fPINNs) constitute a methodological framework designed to address forward and inverse problems for time-fractional diffusion equations by leveraging the Laplace transform. This approach reformulates the original partial differential equation (PDE) with a Caputo time-fractional derivative into the Laplace (complex frequency) domain, allowing standard neural network architectures to approximate solutions without the obstacles of time-history dependence or nonlocal derivatives. Laplace-fPINNs have been demonstrated to solve subdiffusion problems efficiently and accurately in both low and high spatial dimensions, as well as to recover spatially-varying parameters from sparse observations (Yan et al., 2023).

1. Time-Fractional Subdiffusion and Laplace Transform Framework

Laplace-fPINNs are primarily formulated for initial-boundary value problems for subdiffusion, modeled by the Caputo time-fractional diffusion equation: $$\partial_{0+}^\alpha u(x,t) = \nabla\cdot\bigl(a(x)\nabla u(x,t)\bigr) + c(x)u(x,t) + f(x,t),$$ where $x\in\Omega\subset\mathbb{R}^d$, $t\in (0,T]$, $0<\alpha<1$, and $\partial_{0+}^\alpha$ denotes the Caputo derivative: $$\partial_{0+}^{\alpha}u(x, t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-\tau)^{-\alpha}\ \frac{\partial u(x, \tau)}{\partial \tau}d\tau.$$

Applying the Laplace transform in time transforms the nonlocal Caputo derivative into a local algebraic term: $$\mathcal{L}\{\partial_{0+}^\alpha u\}(s) = s^\alpha \tilde u(x, s) - s^{\alpha-1}u_0(x).$$ The Laplace-domain PDE becomes: $$s^\alpha \tilde u(x, s) - \nabla \cdot (a(x)\nabla \tilde u(x, s)) - c(x)\tilde u(x, s) = s^{\alpha-1}u_0(x) + \tilde f(x,s),$$ subject to transformed boundary conditions.

2. Laplace-Domain Neural Network and Loss Function

The Laplace-fPINN parameterizes the Laplace-domain solution via a feed-forward neural network $\tilde u_{NN}(x, s;\theta)$, taking as input $(x, s) \in \Omega\times[S_{\min}, S_{\max}]$. The PDE operator in Laplace space is defined as: $x\in\Omega\subset\mathbb{R}^d$0 with right-hand side $x\in\Omega\subset\mathbb{R}^d$1. The physics residual is

$x\in\Omega\subset\mathbb{R}^d$2

The loss function to be minimized is: $x\in\Omega\subset\mathbb{R}^d$3 where \begin{align*} L_{\rm eq}^{\rm lp}(\theta) &= \frac{1}{N_r} \sum_{i=1}^{N_r} |r_{NN}(x_r^{(i)}, s_r^{(i)}; \theta)|^2, \ L_{\rm bd}^{\rm lp}(\theta) &= \frac{1}{N_{bd}} \sum_{j=1}^{N_{bd}} |\tilde u_{NN}(x_{bd}^{(j)}, s_{bd}^{(j)}; \theta)|^2, \end{align*} for sets of interior and boundary collocation points, respectively.

A fundamental advantage is the removal of time-history dependence, eliminating the need for discretization or auxiliary points to evaluate fractional derivatives (Yan et al., 2023).

3. Numerical Inversion of the Laplace Transform

After training, the Laplace-domain network prediction $x\in\Omega\subset\mathbb{R}^d$4 must be inverted to recover the time-domain solution $x\in\Omega\subset\mathbb{R}^d$5. Laplace-fPINNs adopt the Gaver–Stehfest algorithm for this numerical inversion: $x\in\Omega\subset\mathbb{R}^d$6 with $x\in\Omega\subset\mathbb{R}^d$7 an even integer (typically $x\in\Omega\subset\mathbb{R}^d$8 for optimal trade-off), and the weights $x\in\Omega\subset\mathbb{R}^d$9 given by: $t\in (0,T]$0

Comparative analysis of inverse Laplace transform algorithms relevant to Laplace-fPINNs is exhaustively reviewed in (Kuhlman, 2012), including Stehfest, Schapery, Weeks, fixed Talbot, and Fourier-series with de Hoog acceleration. These methods differ in reuse strategies, parameter tuning, and behavior under various signal classes. For Laplace-fPINNs, Stehfest is favored for simplicity when solutions are monotonic and non-oscillatory.

Summary Table: Properties of Inverse Laplace Algorithms

Method	p(t)?	Reuse?	Complex?	Free Params	N per batch	Time Behavior
Stehfest	yes	no	no	none	10–18	smooth; non-oscillatory
Schapery	no	yes	no	$t\in (0,T]$1 (trial-and-error)	5–20	smooth decay to steady
Weeks	no	yes	yes	$t\in (0,T]$2 (sensitive)	20–100	wide; sensitive
Fixed Talbot	no	yes	yes	$t\in (0,T]$3 (automatic)	20–100	smooth decays
Fourier+de Hoog	no	yes	yes	$t\in (0,T]$4 (automatic)	25–100	robust, oscillatory

Algorithms with $t\in (0,T]$5 independent of $t\in (0,T]$6 enable reuse of image function evaluations across multiple time points, critical for efficient Laplace-fPINN deployment in high-cost PDE scenarios (Kuhlman, 2012).

4. Training for Forward and Inverse Subdiffusion

Laplace-fPINNs address both forward and inverse problems. For forward problems with known parameters $t\in (0,T]$7, training minimizes $t\in (0,T]$8, and time-domain recovery is performed post-training by Laplace inversion. For inverse problems where $t\in (0,T]$9 or $0<\alpha<1$0 are unknown, Laplace-fPINNs introduce additional parameterizations, e.g., $0<\alpha<1$1, with expanded loss: $0<\alpha<1$2 where $0<\alpha<1$3 enforces observational fidelity in Laplace space, and $0<\alpha<1$4 applies to known parameter priors or boundary values.

This hierarchical loss structure enables simultaneous recovery of unknown parameters and solutions from sparse or noisy data.

5. Numerical Results, Scalability, and Error Metrics

In two-dimensional forward tests, e.g., $0<\alpha<1$5, $0<\alpha<1$6, $0<\alpha<1$7, with low-rank initial and forcing functions, Laplace-fPINNs produced a mean relative $0<\alpha<1$8 error of approximately $0<\alpha<1$9 using a $\partial_{0+}^\alpha$0 neural network with inversion parameter $\partial_{0+}^\alpha$1; higher $\partial_{0+}^\alpha$2 resulted in worse errors due to instability. Increasing network width and depth further reduced error (e.g., $\partial_{0+}^\alpha$3 for $\partial_{0+}^\alpha$4 nets). Three-dimensional forward and inverse subdiffusion benchmarks on manufactured solutions and parameter recovery yielded visually indistinguishable results, achieving pointwise errors below $\partial_{0+}^\alpha$5 in parameter estimation (Yan et al., 2023).

6. Algorithmic and Practical Considerations

Laplace-fPINNs confer several computational and methodological advantages:

• No discretization or quadrature of the Caputo derivative, eliminating dense time-history sampling.
• No auxiliary points for temporal memory, unlike time-domain PINNs for fractional PDEs.
• Computational cost per optimizer iteration is independent of final simulation time $\partial_{0+}^\alpha$6; cost primarily depends on spatial dimension and network width.
• Demonstrated scalability to problems in three or more spatial dimensions.

General recommendations for algorithm selection in Laplace-fPINNs include:

• Precompute image-function $\partial_{0+}^\alpha$7 on a fixed $\partial_{0+}^\alpha$8-grid for batch time points where possible.
• Employ fixed Talbot or Fourier–de Hoog for more robust inversion in the presence of oscillatory or discontinuous time-domain behaviors, especially when batch evaluation is required (Kuhlman, 2012).
• For monotonic and non-oscillatory problems, Stehfest inversion is efficient and straightforward to implement.
• Use complex $\partial_{0+}^\alpha$9-domain methods when fPINN software infrastructure supports complex arithmetic, gaining convergence and batch-evaluation advantages.

7. Broader Context and Methodological Implications

The Laplace-fPINN approach addresses a central challenge in the PINN methodology for anomalous diffusion and memory-driven transport: automatic differentiation is not directly applicable to fractional derivatives in time. By relocating the learning problem to the Laplace domain, where these derivatives become local and algebraic, Laplace-fPINNs align the strengths of neural modeling with established tools for Laplace-space inversion and sparse spatial sampling. The methodology leverages decades of algorithmic development in numerical Laplace inversion, as indexed in (Kuhlman, 2012).

This suggests a plausible direction for further refinement: hybridization of Laplace-fPINNs with advanced inversion schemes and adaptive $$\partial_{0+}^{\alpha}u(x, t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-\tau)^{-\alpha}\ \frac{\partial u(x, \tau)}{\partial \tau}d\tau.$$0-grid selection could enhance performance for problems with rapid transient or multi-scale time dynamics, beyond current Stehfest-centered implementations.