---
title: Laplace-fPINNs for Subdiffusion Problems
url: https://www.emergentmind.com/topics/laplace-transform-methods-laplace-fpinns
type: topic
---

# Laplace-fPINNs for Subdiffusion Problems

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 [2304.00909].

## 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:
\[
\mathcal A[\tilde u](x, s) = s^\alpha \tilde u(x, s) - \nabla\cdot(a(x)\nabla \tilde u(x, s)) - c(x)\tilde u(x, s)
\]
with right-hand side $F(x,s) = s^{\alpha-1}u_0(x) + \tilde f(x, s)$. The physics residual is
\[
r_{NN}(x, s; \theta) = \mathcal A[\tilde u_{NN}](x, s) - F(x, s).
\]

The loss function to be minimized is:
\[
L^{\rm lp}(\theta) = w_{\rm eq} L_{\rm eq}^{\rm lp}(\theta) + w_{\rm bd} L_{\rm bd}^{\rm lp}(\theta),
\]
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 [2304.00909].

## 3. Numerical Inversion of the Laplace Transform

After training, the Laplace-domain network prediction $\tilde u_{NN}(x, s; \theta^*)$ must be inverted to recover the time-domain solution $u_{NN}(x, t; \theta^*)$. Laplace-fPINNs adopt the Gaver–Stehfest algorithm for this numerical inversion:
\[
u_{NN}(x, t; \theta^*) \approx \frac{\ln 2}{t} \sum_{i=1}^{M} \mu_i \tilde u_{NN}(x, s_i; \theta^*), \qquad s_i = \frac{i \ln 2}{t},
\]
with $M$ an even integer (typically $M=4$ for optimal trade-off), and the weights $\mu_i$ given by:
\[
\mu_i = (-1)^{M/2 + i} \sum_{k= \lceil(i+1)/2\rceil}^{\min(i, M/2)} \frac{k^{M/2} (2k)!}{(\frac{M}{2} - k)! k! (k-1)! (i-k)! (2k-i)!}.
\]

Comparative analysis of inverse Laplace transform algorithms relevant to Laplace-fPINNs is exhaustively reviewed in [1204.4754], 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       | $p_i$ (trial-and-error)        | 5–20        | smooth decay to steady  |
| Weeks          | no    | yes    | yes      | $\kappa, b$ (sensitive)        | 20–100      | wide; sensitive         |
| Fixed Talbot   | no    | yes    | yes      | $r=2N/(5t_{max})$ (automatic)  | 20–100      | smooth decays           |
| Fourier+de Hoog| no    | yes    | yes      | $\gamma,T,\epsilon$ (automatic)| 25–100      | robust, oscillatory     |

Algorithms with $p_j$ independent of $t$ enable reuse of image function evaluations across multiple time points, critical for efficient Laplace-fPINN deployment in high-cost PDE scenarios [1204.4754].

## 4. Training for Forward and Inverse Subdiffusion

Laplace-fPINNs address both forward and inverse problems. For forward problems with known parameters $\alpha, a, c$, training minimizes $L^{\rm lp}(\theta)$, and time-domain recovery is performed post-training by Laplace inversion. For inverse problems where $a(x)$ or $\alpha$ are unknown, Laplace-fPINNs introduce additional parameterizations, e.g., $a(x) \approx a_{NN}(x; \theta_2)$, with expanded loss:
\[
L^{\rm lp}(\theta_1, \theta_2) = w_{\rm eq}L_{\rm eq}^{\rm lp} + w_{\rm bd}L_{\rm bd}^{\rm lp} + w_{\rm obs}L_{\rm obs}^{\rm lp} + w_{\rm prior}L_{\rm prior}^{\rm lp},
\]
where $L_{\rm obs}^{\rm lp}$ enforces observational fidelity in Laplace space, and $L_{\rm prior}^{\rm lp}$ 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., $\Omega = [0,1]^2$, $T=1$, $\alpha \approx 0.5$, with low-rank initial and forcing functions, Laplace-fPINNs produced a mean relative $\ell_2$ error of approximately $3.48\times10^{-2}$ using a $5\times256$ neural network with inversion parameter $M=4$; higher $M$ resulted in worse errors due to instability. Increasing network width and depth further reduced error (e.g., $\sim 2.04\times10^{-2}$ for $512\times7$ nets). Three-dimensional forward and inverse subdiffusion benchmarks on manufactured solutions and parameter recovery yielded visually indistinguishable results, achieving pointwise errors below $10^{-2}$ in parameter estimation [2304.00909].

## 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 $T$; 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 $\bar f(p)$ on a fixed $p$-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 [1204.4754].
- For monotonic and non-oscillatory problems, Stehfest inversion is efficient and straightforward to implement.
- Use complex $p$-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 [1204.4754].

*This suggests* a plausible direction for further refinement: hybridization of Laplace-fPINNs with advanced inversion schemes and adaptive $p$-grid selection could enhance performance for problems with rapid transient or multi-scale time dynamics, beyond current Stehfest-centered implementations.

Source: https://www.emergentmind.com/topics/laplace-transform-methods-laplace-fpinns