Learnable Fractional Reaction-Diffusion (LFRD2)

• The paper introduces a hybrid framework that embeds a learnable time-fractional reaction-diffusion PDE for iterative refinement of initial depth estimates.
• It leverages fractional calculus to capture long-memory effects and nonlocal diffusion, significantly enhancing under-display Time-of-Flight image restoration.
• Empirical results demonstrate that adaptively learning the fractional order and continuous convolution operator outperforms traditional methods with minimal extra parameters.

Searching arXiv for LFRD2 and closely related fractional reaction–diffusion work. Learnable Fractional Reaction-Diffusion Dynamics (LFRD2) denotes a hybrid, physics-informed framework that embeds a time-fractional reaction-diffusion evolution law inside a learnable depth-restoration pipeline. In its defining formulation, introduced for under-display Time-of-Flight imaging, LFRD2 combines a deep initial estimator with a Deep Fractional Reaction-Diffusion module that performs iterative refinement through a Caputo time-fractional reaction-diffusion update with learnable differential orders and an efficient continuous convolution operator (Qiao et al., 3 Nov 2025). More broadly, LFRD2 sits at the intersection of fractional PDE modeling, operator-based image restoration, and learnable nonlocal dynamics: it inherits long-memory temporal behavior from fractional calculus, nonlocal transport structure from fractional diffusion theory, and data adaptivity from neural parameterization [(Qiao et al., 3 Nov 2025); (Saxena et al., 2015); (Saxena et al., 2012)].

1. Origin and defining formulation

The term LFRD2 originates in “Learnable Fractional Reaction-Diffusion Dynamics for Under-Display ToF Imaging and Beyond” (Qiao et al., 3 Nov 2025). The framework was introduced to address under-display ToF degradation caused by transparent OLED layers, specifically signal attenuation, multi-path interference, and temporal noise, which significantly compromise depth quality (Qiao et al., 3 Nov 2025). The proposed remedy is a two-stage architecture composed of a Deep Initial State Builder and a Deep Fractional Reaction-Diffusion module (Qiao et al., 3 Nov 2025).

In this formulation, the initial depth estimate $u_0$ is produced by a backbone network, and refinement proceeds iteratively through

$$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$

where $\mathcal{D}_t$ is a diffusion term, $\mathcal{R}_t$ is a reaction term, and $w_t$ are memory weights induced by the fractional derivative (Qiao et al., 3 Nov 2025). The underlying PDE is written as

$$^C_0 D_t^{\alpha} u_{n+1} = \mathrm{div}\bigl(g(|\nabla u_n|)\nabla u_n\bigr) + \lambda (u_0 - u_n),$$

with $^C_0 D_t^{\alpha}$ the Caputo time-fractional derivative, $g(\cdot)$ a learnable diffusivity, and $\lambda=0.01$ following TNRD (Qiao et al., 3 Nov 2025).

This structure places LFRD2 in a longer lineage of fractional reaction-diffusion models. Earlier work studied computable solutions for fractional reaction-diffusion equations with generalized Riemann-Liouville time derivatives and Riesz-Feller space derivatives, using Laplace-Fourier methods and generalized Mittag-Leffler functions (Saxena et al., 2015). Related work also derived closed-form solutions for distributed-order systems with Riemann-Liouville time derivatives and Riesz-Feller space operators, with solutions expressed through Mittag-Leffler and Srivastava-Daoust functions (Saxena et al., 2012). LFRD2 differs in that it does not primarily seek closed-form PDE solutions; instead, it makes the PDE dynamics learnable and task-adaptive (Qiao et al., 3 Nov 2025).

2. Fractional temporal memory and reaction-diffusion structure

The central mathematical distinction of LFRD2 is the replacement of the standard first-order time derivative by a Caputo derivative of order $\alpha\in(0,1)$, defined as

$$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$0

This kernel weights all past times, so the refinement at iteration $$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$1 depends on the entire history of previous states rather than only on $$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$2 (Qiao et al., 3 Nov 2025). The paper explicitly contrasts this with integer-order dynamics, where the update is Markovian (Qiao et al., 3 Nov 2025).

The diffusion term in LFRD2 has Perona-Malik form,

$$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$3

while the reaction term is

$$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$4

The reaction component anchors the evolution to the initial estimate, thereby counteracting excessive smoothing, whereas the diffusion component performs nonlinear spatial regularization (Qiao et al., 3 Nov 2025). This split is consistent with broader fractional reaction-diffusion literature, where diffusion is represented by nonlocal or fractional operators and reaction contributes local decay, growth, or forcing terms (Saxena et al., 2015, Zhang et al., 2019, Wang et al., 2018).

A key conceptual point is that fractional time derivatives encode memory through power-law kernels rather than exponential forgetting. Earlier analytical studies showed that time-fractional reaction and diffusion equations are naturally expressed in terms of generalized Mittag-Leffler functions, which generalize exponential relaxation and yield heavy-tailed temporal behavior [(Saxena et al., 2010); (Saxena et al., 2015); (Saxena et al., 2012)]. This suggests that LFRD2’s use of a learnable $$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$5 is not merely a numerical variation on iterative refinement; it situates the model within a class of non-Markovian evolution laws whose effective memory spectrum can be adapted from data (Qiao et al., 3 Nov 2025).

3. Discretization, learnability, and continuous convolution

LFRD2 discretizes the Caputo derivative with the L1 approximation

$$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$6

where

$$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$7

With $$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$8, this leads to the explicit update

$$u_{n+1} = \sum_{t=0}^n w_t u_t + \mathcal{D}_t(u_n) + \mathcal{R}_t(u_n, u_0),$$9

with

$\mathcal{D}_t$0

The second line is the fractional history correction, and it is the mechanism by which all previous increments influence the current refinement (Qiao et al., 3 Nov 2025).

The differentiability order $\mathcal{D}_t$1 is not fixed. The paper states that the differentiation orders are dynamically generated by a neural network, and ablations comparing fixed $\mathcal{D}_t$2 against the variable-order model show that the variable-order model performs best (Qiao et al., 3 Nov 2025). This is the defining “learnable” aspect of LFRD2: the model does not merely tune coefficients in a preset PDE but adapts the effective temporal memory law itself (Qiao et al., 3 Nov 2025).

The spatial diffusion operator is implemented through a continuous convolution construction rather than a standard finite-difference stencil. Building on a representation of convolution via repeated integration and differentiation,

$\mathcal{D}_t$3

LFRD2 predicts estimated Dirac-delta-like kernel quantities directly from network features and approximates the repeated antiderivative by a local linear form $\mathcal{D}_t$4 (Qiao et al., 3 Nov 2025). The resulting operator is designed to be more efficient than Neural Field Convolutions while preserving a continuous-operator interpretation (Qiao et al., 3 Nov 2025).

This learnable-operator view has clear precedents in the fractional PDE literature. Spectral formulations of the fractional Laplacian show that in Fourier space the operator acts by multiplication with $\mathcal{D}_t$5 on periodic domains (Zhang et al., 2019). Riesz and Riesz-Feller derivatives similarly admit explicit Fourier symbols such as $\mathcal{D}_t$6 in the symmetric case or $\mathcal{D}_t$7 in the skew case [(Saxena et al., 2015); (Saxena et al., 2012)]. LFRD2 does not explicitly parameterize these classical symbols, but it occupies the same operator-theoretic niche: a learnable approximation to nonlocal smoothing constrained by PDE form.

4. Relation to broader fractional reaction-diffusion theory

Fractional reaction-diffusion systems predate LFRD2 by many years. Earlier analytical work established explicit solution formulas for time-fractional and space-fractional equations using Laplace and Fourier transforms, with generalized Mittag-Leffler and Fox $\mathcal{D}_t$8-function kernels [(Saxena et al., 2010); (Saxena et al., 2015); (Saxena et al., 2012)]. These studies make precise how fractional orders control memory, anomalous transport, and relaxation tails.

Distributed-order and multi-term time operators are especially relevant. One line of work considered equations of the form

$\mathcal{D}_t$9

with generalized Riemann-Liouville time derivatives and Riesz-Feller space derivatives, interpreting the two-term time operator as a discrete distributed-order model (Saxena et al., 2015). Another line studied

$\mathcal{R}_t$0

and derived closed forms through Prabhakar and Srivastava-Daoust functions (Saxena et al., 2012). These results show that learnable fractional dynamics can, in principle, target not only a single time order but a mixture or distribution of temporal exponents.

The stochastic derivation of fractional transport from continuous-time random walks further clarifies the modeling role of these operators. Heavy-tailed waiting times lead to Caputo time derivatives, while heavy-tailed jumps lead to Riesz space derivatives, yielding equations such as

$\mathcal{R}_t$1

or fractional reaction-diffusion equations where the reaction term itself is convolved with the same memory kernel (Verbeeck et al., 17 Mar 2025). This provides a mechanistic interpretation of why a learnable $\mathcal{R}_t$2 can be meaningful: it may encode effective waiting-time or transport heterogeneity rather than serving only as a generic tuning parameter (Verbeeck et al., 17 Mar 2025).

On bounded domains, regularity theory shows that solution smoothness is limited by endpoint singular behavior determined by the fractional operator parameters (Ervin, 2019). This suggests a practical implication for learnable models: naive Euclidean smoothness assumptions may be mismatched to the actual function spaces of fractional PDE solutions. A plausible implication is that LFRD2-style models could benefit from architecture or loss designs that better reflect weighted Sobolev or nonlocal regularity structures, especially outside image-restoration settings (Ervin, 2019).

5. Stability, well-posedness, and numerical dynamics

Fractional reaction-diffusion systems require a distinct stability theory because temporal decay is governed by sectorial spectral conditions rather than only by $\mathcal{R}_t$3. Recent work proved a linearization principle for abstract fractional reaction-diffusion equations with Caputo derivative of order $\mathcal{R}_t$4, showing that stability is associated with the spectral condition $\mathcal{R}_t$5, while instability arises when some eigenvalue satisfies $\mathcal{R}_t$6 (Ahmad et al., 2 Jul 2025). The same work established a counterpart of classical Turing instability for fractional systems (Ahmad et al., 2 Jul 2025). This provides a rigorous backdrop for any learnable fractional dynamics whose iterates approximate a time-fractional PDE.

Well-posedness on bounded domains with spectral fractional Laplacians has also been analyzed using analytic semigroup theory. For semilinear problems of the form

$\mathcal{R}_t$7

one can reformulate inhomogeneous boundary conditions via harmonic lifting, derive Duhamel formulas, obtain local wellposedness for locally Lipschitz reactions, and establish positivity preservation and invariant sets for specific systems such as bistable equations and Gray-Scott dynamics (Yuan et al., 29 Jan 2026). This is relevant because it shows that learnable reaction terms can, at least in principle, be constrained so that the overall fractional dynamics remain well posed and positivity preserving (Yuan et al., 29 Jan 2026).

Numerical studies further clarify what fractional order changes in practice. Stabilized semi-implicit Fourier spectral schemes for nonlinear space-fractional reaction-diffusion systems have shown that varying $\mathcal{R}_t$8 changes morphology in fractional Allen-Cahn, Gray-Scott, and FitzHugh-Nagumo models, and that the effect of changing $\mathcal{R}_t$9 is not equivalent to simply rescaling the diffusion coefficient (Zhang et al., 2019). Related Gray-Scott simulations on bounded domains showed that decreasing $w_t$0 yields different steady patterns and a scaling law for radial distribution functions in terms of fractional order (Wang et al., 2018). These results directly rebut a common oversimplification: fractional order is not merely an alternative parameterization of diffusion strength. It changes the operator class and therefore the spatiotemporal dynamics themselves (Zhang et al., 2019, Wang et al., 2018).

LFRD2’s empirical design choices are consistent with this broader picture. The model uses an explicit fractional update with learned order, reaction anchoring, and continuous convolution, and the paper reports that training can become unstable with poor strategies and that future work may consider implicit schemes (Qiao et al., 3 Nov 2025). This suggests that stability-aware numerical design remains an open issue for learnable fractional PDE modules, even when their task-level performance is strong.

6. Applications, empirical performance, and scope

LFRD2 was introduced for under-display ToF imaging, but its empirical scope is broader. On four benchmark datasets, the paper reports effectiveness for under-display ToF restoration, generic ToF denoising, and depth super-resolution (Qiao et al., 3 Nov 2025). On SUD-ToF, LFRD2 achieved MAE $w_t$1 mm and RMSE $w_t$2 mm, improving over UD-ToFnet at MAE $w_t$3 mm and RMSE $w_t$4 mm (Qiao et al., 3 Nov 2025). On RUD-ToF, it achieved MAE $w_t$5 mm and RMSE $w_t$6 mm, compared with UD-ToFnet at MAE $w_t$7 mm and RMSE $w_t$8 mm (Qiao et al., 3 Nov 2025). On FLAT, it achieved MAE $w_t$9 mm and RMSE $$^C_0 D_t^{\alpha} u_{n+1} = \mathrm{div}\bigl(g(|\nabla u_n|)\nabla u_n\bigr) + \lambda (u_0 - u_n),$$0 mm (Qiao et al., 3 Nov 2025). On NYUv2, it improved over DSR-EI across $$^C_0 D_t^{\alpha} u_{n+1} = \mathrm{div}\bigl(g(|\nabla u_n|)\nabla u_n\bigr) + \lambda (u_0 - u_n),$$1, $$^C_0 D_t^{\alpha} u_{n+1} = \mathrm{div}\bigl(g(|\nabla u_n|)\nabla u_n\bigr) + \lambda (u_0 - u_n),$$2, and $$^C_0 D_t^{\alpha} u_{n+1} = \mathrm{div}\bigl(g(|\nabla u_n|)\nabla u_n\bigr) + \lambda (u_0 - u_n),$$3 upsampling factors, reporting $$^C_0 D_t^{\alpha} u_{n+1} = \mathrm{div}\bigl(g(|\nabla u_n|)\nabla u_n\bigr) + \lambda (u_0 - u_n),$$4, $$^C_0 D_t^{\alpha} u_{n+1} = \mathrm{div}\bigl(g(|\nabla u_n|)\nabla u_n\bigr) + \lambda (u_0 - u_n),$$5, and $$^C_0 D_t^{\alpha} u_{n+1} = \mathrm{div}\bigl(g(|\nabla u_n|)\nabla u_n\bigr) + \lambda (u_0 - u_n),$$6 in MAE/RMSE, respectively (Qiao et al., 3 Nov 2025).

The ablations are especially revealing. Removing fractional calculus degrades performance, as does removing continuous convolution, and a variable-order model outperforms fixed-order alternatives (Qiao et al., 3 Nov 2025). Comparisons with GRU- and LSTM-based iterative refinements also show that the fractional dynamics module achieves the best MAE/RMSE trade-off among the tested recurrent refinements while adding only $$^C_0 D_t^{\alpha} u_{n+1} = \mathrm{div}\bigl(g(|\nabla u_n|)\nabla u_n\bigr) + \lambda (u_0 - u_n),$$7M parameters over the UD-ToFnet backbone (Qiao et al., 3 Nov 2025). These results support the claim that the benefit is not reducible to generic recurrence alone.

A broader implication is that LFRD2 should be viewed less as a ToF-specific architecture than as a learnable PDE layer for inverse problems where one has an initial estimate and benefits from memory-aware iterative refinement. The paper itself makes this point by replacing the initial-state builder according to task and using the same fractional refinement mechanism “for under-display ToF imaging and beyond” (Qiao et al., 3 Nov 2025). This suggests that the core transferable ingredient is the learnable fractional reaction-diffusion update, not the upstream sensor-specific encoder.

A common misconception is that LFRD2 is simply a classical reaction-diffusion network with one extra hyperparameter. The surrounding literature indicates otherwise. Fractional derivatives alter the temporal operator from local to nonlocal, fractional spatial operators alter transport from Gaussian to heavy-tailed or skewed propagation, and distributed-order variants introduce multi-scale memory unavailable to single-order models [(Saxena et al., 2010); (Saxena et al., 2015); (Saxena et al., 2012); (Verbeeck et al., 17 Mar 2025)]. LFRD2 adopts only part of this full fractional toolkit, but it does so in a way that makes the memory law itself learnable (Qiao et al., 3 Nov 2025).

Another misconception is that learnability necessarily sacrifices interpretability. In LFRD2 the neural components parameterize diffusivity, order selection, and convolution kernels, but the update law remains an explicit discretization of a fractional reaction-diffusion PDE (Qiao et al., 3 Nov 2025). This does not make the model analytically transparent in the same sense as closed-form Mittag-Leffler solutions, but it preserves a strong operator-level interpretation absent from purely black-box regressors.

The main unresolved issues concern stability, parameter identifiability, and extension beyond the present formulation. The broader literature points to several natural directions: richer distributed-order time operators [(Saxena et al., 2015); (Saxena et al., 2012)], explicit stability regularization based on fractional linearization theory (Ahmad et al., 2 Jul 2025), better bounded-domain operator design informed by well-posedness and invariant sets (Yuan et al., 29 Jan 2026), and architectures adapted to the limited regularity of fractional PDE solutions on bounded domains (Ervin, 2019). Within that larger landscape, LFRD2 can be understood as an initial, application-driven instance of a more general program: making fractional reaction-diffusion dynamics trainable while preserving enough PDE structure to remain physically interpretable (Qiao et al., 3 Nov 2025).