fPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations

Binghang Lu; Christian Moya; Guang Lin; Zhaopeng Hao

arxiv: 2605.16594 · v1 · pith:CUQJZBOAnew · submitted 2026-05-15 · 🧮 math.NA · cs.LG· cs.NA

fPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations

Binghang Lu , Zhaopeng Hao , Christian Moya , Guang Lin This is my paper

Pith reviewed 2026-05-19 21:13 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NA

keywords fractional partial differential equationsoperator learningphysics-informed neural networksDeepONetCaputo derivativediffusion-wave equationsnumerical approximationvariable order

0 comments

The pith

The fPINN-DeepONet framework solves multi-term time-fractional mixed diffusion-wave equations with variable fractional orders by combining operator learning and an L2 approximation for Caputo derivatives.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a physics-informed deep operator learning method for multi-term time-fractional mixed diffusion-wave equations that handles both fixed and dynamically varying fractional orders in space and time. The authors derive an L2 approximation for the Caputo fractional derivative of order between 1 and 2 that reaches first-order accuracy, then embed the approximation inside a DeepONet architecture to learn the solution operator while enforcing the fractional physics. Numerical tests show the resulting framework stays accurate and stable even when the data contain noise. A sympathetic reader would care because these equations describe anomalous transport and wave behavior in viscoelastic or porous media, where standard solvers become slow or unstable once orders vary or measurements are imperfect.

Core claim

We develop the fPINN-DeepONet by deriving an L2 approximation which achieves first-order accuracy for the Caputo fractional derivative of order β ∈ (1,2). This approximation is integrated into the operator learning framework to solve the equations, and numerical experiments confirm its accuracy, robustness to noise, and ability to handle space-time varying fractional orders.

What carries the argument

fPINN-DeepONet, which embeds the derived L2 approximation for Caputo derivatives inside a physics-informed DeepONet to learn solution operators for fractional PDEs.

If this is right

The framework solves multi-term time-fractional mixed diffusion-wave equations.
It handles dynamically varying fractional orders in both space and time.
It remains accurate and robust when input data contain noise.
It applies equally to fixed-order and variable-order versions of the target equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same embedding of the L2 scheme could be tested on other fractional operators such as Riemann-Liouville derivatives.
Once the operator is learned, repeated queries for new initial conditions or source terms become cheap, suggesting use in real-time control of anomalous-diffusion systems.
Scaling experiments in three or more spatial dimensions would test whether the current training cost grows acceptably with problem size.

Load-bearing premise

The derived L2 approximation achieves first-order accuracy for the Caputo fractional derivative of order β ∈ (1,2) and integrates directly into the operator learning architecture without loss of stability or consistency.

What would settle it

Run the L2 approximation on a test function with known Caputo derivative of order 1.5 and verify that the error decreases linearly with the time-step size; any deviation from first-order convergence would falsify the accuracy claim.

Figures

Figures reproduced from arXiv: 2605.16594 by Binghang Lu, Christian Moya, Guang Lin, Zhaopeng Hao.

**Figure 2.** Figure 2: Fixed Fractional Order FODE Forward Problem: Comparison of the exact solution and [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: Fixed Fractional Order FPDE Forward Problem: The subfigures (from left to right) [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: One-Dimensional Two-Term FPDE Forward Problem: The subfigures (from left to right) [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Variable Fractional Order FODE Forward Problem: The subfigure on the left shows the [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: Variable Fractional Order FODE Inverse Problem: The subfigure on the left shows the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: FPDE Forward Problem: The subfigures (from left to right) display the exact solution, [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: FPDE Inverse Problem: The subfigure on the left shows the training loss as a function [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

**Figure 9.** Figure 9: α(x, t) FPDE Forward Problem: The subfigures (from left to right) display the exact solution, the fPINN-DeepONet prediction, and the corresponding absolute error. As shown in [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗

**Figure 10.** Figure 10: α(x, t) FPDE Inverse Problem: The subfigures (from left to right) show the exact fractional order, the fPINN-DeepONet prediction, and the corresponding absolute error. 4.5 High-dimensional FPDE In this section, we extended the spatial domain to multiple dimensions, e.g., (x1, x2, . . . xn, t) instead of just x, t by introducing another spatial variable xi ∈ [0, π]. The equation can be shown as: C 0 D α(t)… view at source ↗

**Figure 11.** Figure 11: Heatmap of testing error of high-dimensional FPDE at [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗

**Figure 12.** Figure 12: Subfigures (from left to right) display the exact solution, the fPINN-DeepONet predic [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗

**Figure 13.** Figure 13: Discontinuous fractional order Inverse Problem: A comparison between the true frac [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗

**Figure 14.** Figure 14: Hyperparameter Testing: The subfigure on the left illustrates test error varies with [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗

read the original abstract

In this paper, we develop a physics-informed deep operator learning framework for solving multi-term time-fractional mixed diffusion-wave equations (TFMDWEs). We begin by deriving an $L_2$ approximation, which achieves first-order accuracy for the Caputo fractional derivative of order $\beta \in (1,2)$. Building upon this foundation, we propose the fPINN-DeepONet framework, a novel approach that integrates operator learning with the $L_2$ approximation to efficiently solve fractional partial differential equations (FPDEs). Our framework is successfully applied to both fixed and variable fractional-order PDEs, demonstrating the framework's versatility and broad applicability. To evaluate the performance of the proposed model, we conduct a series of numerical experiments that involve dynamically varying fractional orders in both space and time, as well as scenarios with noisy data. These results highlight the accuracy, robustness, and efficiency of the fPINN-DeepONet framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper combines an L2 Caputo approximation with DeepONet to handle multi-term variable-order time-fractional diffusion-wave equations, but the abstract supplies no error tables or baselines to back the accuracy claims.

read the letter

The main point is that they derive a first-order L2 scheme for the Caputo derivative when the order sits between 1 and 2, then fold it into a DeepONet loss to solve multi-term TFMDWEs that allow the fractional order to change in both space and time. They also run tests with noisy data. That combination is the concrete step forward from earlier PINN and DeepONet work on fractional problems. The framework is applied to both fixed-order and variable-order cases, which shows some versatility for memory-dependent models that show up in physics and engineering. The derivation of the L2 approximation itself looks like a reasonable building block, and putting it inside the operator-learning residual is a direct way to avoid discretizing the whole history at every step. On the weak side, the abstract gives no numbers at all—no L2 errors, no convergence rates, no comparison against other fractional solvers or standard DeepONet baselines. That leaves the robustness claims hard to judge from the summary alone. The stress-test concern about variable orders also lands: the usual L2 expansion assumes fixed beta, so when beta(x,t) varies there can be extra commutator terms that are not automatically O(Delta t). If those are not controlled in the residual, the actual accuracy could drop below the stated first-order rate, especially in the noisy experiments. I would want to see the explicit handling of those terms and the actual error tables before accepting the performance story. This is for people who already work on scientific machine learning for anomalous transport or viscoelasticity and need a tool that can take variable orders without a full finite-difference overhaul. A reader who wants a ready-to-use operator network for these equations would get practical value from the numerical section. It is worth sending to peer review; the idea is grounded enough and the target equations are relevant, even if the validation needs tightening.

Referee Report

2 major / 2 minor

Summary. The paper develops the fPINN-DeepONet framework for multi-term time-fractional mixed diffusion-wave equations. It derives an L2 approximation to the Caputo derivative of order β ∈ (1,2) that is claimed to be first-order accurate, then embeds this approximation inside a physics-informed DeepONet architecture to handle both fixed-order and variable-order (space- and time-dependent) fractional PDEs. Numerical experiments are reported for dynamically varying fractional orders and for noisy data, with the abstract asserting that the resulting method is accurate, robust, and efficient.

Significance. If the central consistency claim holds, the work would supply a practical operator-learning route to variable-order fractional PDEs that avoids repeated re-derivation of discretizations when the order function changes. The combination of an explicit L2 history integral with DeepONet branch-trunk structure could be useful for parametric or inverse problems in anomalous transport. However, the absence of quantitative error tables, convergence rates, or baseline comparisons in the provided text leaves the practical gain over existing PINN or DeepONet treatments of fractional operators unquantified.

major comments (2)

[Abstract] Abstract: the central claim that the derived L2 approximation integrates into the DeepONet loss without degrading consistency or stability when fractional orders vary in both space and time is not supported by any displayed error analysis or residual bound. Standard L2 schemes for fixed β rely on uniform time-step assumptions; the variable-order case introduces commutator terms between β(x,t) and the history integral that are not automatically O(Δt) and must be controlled explicitly in the physics-informed residual.
[Numerical experiments] The manuscript supplies no quantitative error tables, baseline comparisons against other fractional PINN or DeepONet variants, or observed convergence rates under mesh refinement. Without these data the assertions of first-order accuracy for the L2 scheme and of robustness under noise cannot be verified from the given text.

minor comments (2)

[Section 3] Notation for the multi-term operator and the precise definition of the L2 quadrature weights should be stated explicitly before the DeepONet loss is written.
[Figures] Figure captions should include the exact values of the noise level σ and the range of the variable order β(x,t) used in each experiment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment point by point below and indicate the revisions we will make to strengthen the presentation of the error analysis and numerical results.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the derived L2 approximation integrates into the DeepONet loss without degrading consistency or stability when fractional orders vary in both space and time is not supported by any displayed error analysis or residual bound. Standard L2 schemes for fixed β rely on uniform time-step assumptions; the variable-order case introduces commutator terms between β(x,t) and the history integral that are not automatically O(Δt) and must be controlled explicitly in the physics-informed residual.

Authors: We agree that a rigorous treatment of the commutator terms is necessary for the variable-order case. The manuscript derives the L2 approximation for fixed β ∈ (1,2) and then applies it within the DeepONet framework by allowing β to be a space-time dependent input to the operator. However, the current text does not explicitly bound the additional terms that arise when β(x,t) varies. In the revised manuscript we will add a dedicated error analysis subsection that derives the consistency error for the variable-order L2 scheme, showing that the commutator contribution remains O(Δt) under standard smoothness assumptions on β, and we will incorporate the corresponding residual bound into the physics-informed loss discussion. revision: yes
Referee: [Numerical experiments] The manuscript supplies no quantitative error tables, baseline comparisons against other fractional PINN or DeepONet variants, or observed convergence rates under mesh refinement. Without these data the assertions of first-order accuracy for the L2 scheme and of robustness under noise cannot be verified from the given text.

Authors: We acknowledge that the numerical section currently emphasizes visual results and qualitative statements rather than tabulated metrics. In the revision we will add tables reporting L² and L^∞ errors for representative fixed- and variable-order problems, observed convergence rates under successive time-step refinement to confirm the first-order accuracy of the L2 scheme, and direct comparisons against a standard fractional PINN baseline and a non-physics-informed DeepONet. We will also include quantitative error statistics for the noisy-data experiments to substantiate the robustness claims. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on independently derived L2 approximation

full rationale

The provided abstract and description state that the authors derive an L2 approximation achieving first-order accuracy for the Caputo derivative at fixed β ∈ (1,2), then integrate this approximation into the fPINN-DeepONet operator-learning architecture. Numerical experiments test the combined framework on variable-order and noisy cases. No equations, claims, or self-citations in the given text reduce reported performance or consistency to quantities defined by the same fitted parameters or prior self-referential results. The central claim remains independent of its inputs by construction, consistent with the reader's assessment of score 2.0 and the absence of any load-bearing self-definition, fitted-input-as-prediction, or ansatz-smuggling patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the accuracy of the newly derived L2 approximation and on the assumption that DeepONet can stably incorporate the fractional operator without additional regularization or architectural changes.

free parameters (1)

fractional order β
β is allowed to vary in space and time; specific functional forms or discretization choices for β are not detailed in the abstract.

axioms (1)

domain assumption The L2 approximation is first-order accurate for the Caputo derivative when β ∈ (1,2).
This accuracy statement is presented as the foundation upon which the operator-learning framework is built.

pith-pipeline@v0.9.0 · 5715 in / 1294 out tokens · 48416 ms · 2026-05-19T21:13:24.672834+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We begin by deriving an L2 approximation, which achieves first-order accuracy for the Caputo fractional derivative of order β ∈ (1,2). ... fPINN-DeepONet framework ... dynamically varying fractional orders in both space and time
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Use the L2 approximation to solve u_approx for f_approx using Eqs. (2.1) and (2.2)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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L. Guo, H. Wu, X. Yu, and T. Zhou, ”Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations,”Computer Methods in Applied Mechanics and Engineering, vol. 400, p. 115523, 2022

work page 2022

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work page internal anchor Pith review Pith/arXiv arXiv 1910

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