🔢 mathematics

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

This paper introduces fPINN-DeepONet, a novel physics-informed deep operator learning framework that combines an $L_2$ approximation for Caputo fractional derivatives with operator learning to accurately and robustly solve multi-term time-fractional mixed diffusion-wave equations with both fixed and variable fractional orders, even in the presence of noisy data.

Original authors: Binghang Lu, Zhaopeng Hao, Christian Moya, Guang Lin

Published 2026-05-19

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Binghang Lu, Zhaopeng Hao, Christian Moya, Guang Lin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a drop of ink spreads through a glass of water. In the real world, this isn't always a smooth, predictable process. Sometimes the ink gets stuck on tiny particles, moves in bursts, or slows down in weird ways that standard physics equations can't quite capture. Scientists call these "anomalous diffusion" processes, and they use special, complex math called fractional equations to describe them.

The problem is, these fractional equations are incredibly hard to solve. Traditional computer methods are like trying to count every single grain of sand on a beach to predict the tide—it takes forever and requires massive computing power.

This paper introduces a new, smarter way to solve these problems using a type of artificial intelligence called fPINN-DeepONet. Here is how it works, broken down into simple concepts:

1. The "Translator" (The Core Idea)

Think of the problem as a language barrier. You have a set of rules (the physics of the equation) and a specific scenario (like a specific starting temperature or a specific "friction" level in the fluid). Usually, you have to run a heavy calculation from scratch every time you change the scenario.

The authors built a Deep Operator Network (DeepONet). Imagine this as a super-smart translator. Instead of learning to translate one specific sentence at a time, it learns the entire language. Once trained, if you give it a new scenario (like a different "friction" level), it instantly predicts the outcome without needing to re-calculate everything from the ground up. It learns the "operator," or the rulebook, rather than just a single answer.

2. The "Physics GPS" (Physics-Informed)

Usually, AI needs millions of examples (like showing a cat picture a million times to recognize a cat). But in science, we don't always have millions of data points.

To fix this, the authors made the AI "physics-informed." They didn't just let the AI guess; they gave it a GPS based on the laws of physics.

The Map: The AI knows the rules of the universe (the equations).
The Check: If the AI's prediction breaks the laws of physics, the GPS screams "Wrong!" and forces the AI to correct itself.
The Result: The AI learns much faster and more accurately because it's guided by the laws of nature, not just random guessing.

3. The "Ruler" (The L2 Approximation)

One of the biggest hurdles in these equations is a tricky math part called the "fractional derivative." It's like trying to measure a curve with a ruler that only has straight edges.

The authors invented a new, precise L2 approximation (think of it as a custom-made, flexible ruler). This ruler allows the AI to measure these tricky curves accurately. They built this ruler directly into the AI's training process, so the AI knows exactly how to handle these complex fractional math problems.

4. What They Tested (The Experiments)

The team put their new "Translator + GPS + Ruler" system through a gauntlet of tests:

Fixed Rules: They solved problems where the "friction" (fractional order) stayed the same. The AI was incredibly accurate.
Changing Rules: They tested scenarios where the "friction" changed over time or space (like a fluid that gets thicker in some spots and thinner in others). The AI handled this smoothly.
The "Guess the Rule" Game (Inverse Problems): Sometimes we know the result (how the ink spread) but don't know the rule (what the friction was). The AI successfully worked backward to figure out the hidden rules.
Noisy Data: They added "static" or noise to the data, like trying to hear a conversation in a loud room. Even with 80% noise, the AI still figured out the answer correctly.
Discontinuous Jumps: They tested a rule that suddenly jumps from one value to another (like a switch flipping). The AI caught these jumps perfectly.

The Bottom Line

The authors created a tool that acts like a universal solver for a very difficult class of physics problems. By combining a new mathematical "ruler" with an AI that learns the rules of physics, they can predict complex diffusion processes quickly and accurately, even when the data is messy or the rules change constantly.

They didn't claim this cures diseases or predicts the stock market in this paper; they simply proved that this new method works exceptionally well for solving these specific, complex mathematical equations that describe how things move and spread in the physical world.

Technical Summary: fPINN-DeepONet for Multi-term Time-Fractional Mixed Diffusion-Wave Equations

Problem Statement
The paper addresses the challenge of solving multi-term time-fractional mixed diffusion-wave equations (TFMDWEs). These equations generalize single-term time-fractional partial differential equations (TFPDEs) and are critical for modeling anomalous diffusion processes, viscoelastic damping, and sub-diffusive motion in velocity fields. The specific problem involves equations of the form:
$\sum_{i=1}^s K_i {}_0^C D_t^{\alpha_i} u(x, t) = \Delta u(x, t) + f(x, t)$
where the fractional orders $\alpha_i$ can range between 0 and 2, and the equation may involve multiple terms with different orders. Traditional numerical methods (finite difference, finite element, spectral methods) struggle with these problems due to the non-local nature of fractional derivatives and the exponential growth in computational cost for high-dimensional or variable-order scenarios. Furthermore, existing deep learning approaches often rely on L1 approximations or focus on single-term equations, leaving a gap in handling multi-term and variable-order fractional PDEs (FPDEs) efficiently.

Methodology
The authors propose fPINN-DeepONet, a physics-informed deep operator learning framework that integrates a novel numerical approximation with the Deep Operator Network (DeepONet) architecture.

L2 Approximation for Fractional Derivatives:
The core numerical contribution is the derivation of an L2 approximation for the Caputo fractional derivative of order $\beta \in (1, 2)$ . This achieves first-order accuracy. The authors combine this with the standard L1 approximation for orders $\alpha \in (0, 1)$ . These approximations are used to discretize the time-fractional derivatives within the loss function, allowing the neural network to compute the physics-based residual without requiring explicit grid-based solvers for the fractional terms.
DeepONet Architecture:
The framework utilizes a DeepONet structure consisting of:
- Branch Net: Encodes the input function. Depending on the problem type, this input can be the source term $f(x,t)$ , the initial conditions, or the fractional order function $\alpha(x,t)$ (for variable-order problems). The input is discretized at a set of sensor points.
- Trunk Net: Processes the query coordinates $(x, t)$ within the spatiotemporal domain.
- Output: The dot product of the Branch and Trunk outputs approximates the solution operator $G^\dagger$ , mapping input functions to the solution $u(x,t)$ .
Loss Function Construction:
The training objective minimizes a total loss function $L = L_{phy} + L_{data}$ :
- Physical Loss ( $L_{phy}$ ): Incorporates the governing FPDE residual, initial conditions, and boundary conditions. Crucially, the fractional derivative terms in the residual are computed using the proposed L2/L1 approximations applied to the network's predicted output.
- Data Loss ( $L_{data}$ ): Measures the mean squared error between the predicted solution and available ground truth data (if available).
Algorithms:
The paper details two specific algorithms:
- Algorithm 1: Solves fixed fractional-order FPDEs where the branch net input is the source term or initial conditions.
- Algorithm 2: Solves variable fractional-order FPDEs where the branch net input is the fractional order function $\alpha(x,t)$ , enabling the model to generalize across different order configurations.

Key Contributions

Novel Approximation Scheme: The derivation and integration of an L2 approximation for Caputo derivatives of order $\beta \in (1, 2)$ into a physics-informed neural network framework.
Operator Learning for Multi-term FPDEs: The first application of DeepONet to multi-term time-fractional mixed diffusion-wave equations, handling both fixed and variable fractional orders within a unified framework.
Versatility in Problem Types: The framework is demonstrated to solve forward problems (predicting solutions), inverse problems (identifying fractional orders), and handle discontinuous fractional orders and high-dimensional spatial domains.
Robustness: The model is tested under noisy data conditions, demonstrating resilience to Gaussian noise levels up to 80%.

Experimental Results
The authors conducted extensive numerical experiments using the PyTorch library:

Fixed Order Problems: The model achieved high accuracy for FODEs and FPDEs with fixed orders (e.g., $\alpha=0.5, \beta=1.5$ ), with relative $L_2$ errors as low as $10^{-6}$ for TFMDWEs.
Variable Order Problems: The framework successfully learned solutions for FPDEs where the fractional order $\alpha$ was a function of time $\alpha(t)$ or space-time $\alpha(x,t)$ . In inverse problems, it accurately recovered the fractional order function with relative errors around $10^{-5}$ .
Discontinuous Orders: The model effectively handled discontinuous fractional orders (step functions), capturing the jumps in the order function with an $L_2$ relative error of 0.0014.
High Dimensions: The approach was extended to 2D spatial domains, demonstrating feasibility for higher-dimensional FPDEs.
Noise Robustness: Even with 20% to 80% Gaussian noise added to the forcing term, the testing error remained stable around $10^{-2}$ , confirming the model's robustness.
Hyperparameter Sensitivity: Experiments showed that test errors decreased with increased data points and optimized network width/depth, achieving errors of $\sim 10^{-5}$ with 200x200 data points and a network width of 150.

Significance and Claims
The paper claims that fPINN-DeepONet provides a versatile and efficient alternative to traditional grid-based methods for solving complex fractional PDEs. By integrating the L2 approximation with operator learning, the framework overcomes the computational bottlenecks associated with the non-local nature of fractional derivatives. The authors emphasize that their approach:

Generalizes well: It handles both fixed and variable fractional orders, including discontinuous cases, without retraining the entire architecture for every new order configuration.
Reduces data requirements: As a physics-informed method, it leverages the governing equations to constrain the solution space, requiring fewer labeled data points compared to purely data-driven approaches.
Remains robust: It maintains accuracy in the presence of significant noise, making it suitable for real-world scenarios where data may be imperfect.

The work positions itself as a step forward in scientific machine learning, specifically addressing the unexplored area of combining L2 approximations with deep operator learning for multi-term and variable-order FPDEs.