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Physics-Informed Deep Neural Operator for Estimation of kinetic parameters in distributed activation energy model for parallel reaction systems

This paper introduces a Physics-Informed Deep Neural Operator (PI-DeepONet) that efficiently and accurately estimates kinetic parameters for multi-step parallel reaction systems by embedding kinetic laws directly into the network architecture, enabling reliable online analysis of complex pyrolysis processes without requiring a priori assumptions.

Original authors: Feng Han, Xueting Shi, Guoliang Peng, Huayang Sun

Published 2026-09-02

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CC BY 4.0

Original authors: Feng Han, Xueting Shi, Guoliang Peng, Huayang Sun

Original paper licensed under CC BY 4.0 (https://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

When scientists want to understand how a material breaks down under heat, they often turn to a technique called thermogravimetric analysis. Imagine placing a tiny sample of a substance on a scale inside a furnace and slowly raising the temperature. As the material heats up, it releases gases and loses weight. By tracking exactly how much mass is lost at every degree, researchers can see the story of the material's decomposition. For simple substances, this story is straightforward, but for complex materials like wood, coal, or plastics, the process is a tangled web of many chemical reactions happening at once. To make sense of this, scientists use mathematical models that try to describe the speed and energy of these reactions. The challenge has long been figuring out the specific numbers that drive these models. Traditional methods often require guessing the shape of the reaction curve beforehand or running slow, repetitive calculations that can get stuck or fail to find the true answer, especially when dozens of reactions are overlapping.

A team of researchers from the Northwest Institute of Nuclear Technology in China has developed a new way to solve this puzzle using a type of artificial intelligence designed to understand the laws of physics. Instead of guessing or grinding through endless calculations, their method learns to read the weight-loss story directly and instantly tell the scientists the hidden numbers that caused it. They applied this approach to a specific model known as the distributed activation energy model, which treats a complex material as if it were made of many tiny, independent parts, each breaking down at its own speed and energy level. The researchers built a neural network, a computer system inspired by the human brain, that was trained to recognize the relationship between the curve of mass loss and the underlying chemical parameters. Crucially, they did not just let the computer guess; they forced the computer to obey the physical laws of chemistry during its training. This means the system learned not just to match patterns, but to understand that the numbers it predicts must actually produce the observed weight loss when plugged back into the equations of reaction.

The team tested this new tool on two different scenarios: a simple case with just one reaction and a more difficult case with two reactions happening simultaneously. In the computer simulations, the system proved remarkably accurate. When given a curve showing how a material lost mass over time, the network could instantly predict the specific energy and speed factors for the reactions with a level of precision that matched the original data almost perfectly. The reconstructed curves, generated by the network's predictions, overlaid the target curves so closely that the difference was barely visible, with statistical measures showing a near-perfect match. This was not just a theoretical exercise; the researchers then took real-world data from a substance called calcium oxalate monohydrate, a material known for breaking down in distinct, separate steps. When they fed the experimental data from this real material into their trained network, the system successfully identified the kinetic parameters for the first two stages of decomposition. The curves of mass loss and reaction speed that the network predicted based on these numbers aligned almost exactly with the actual measurements taken in the lab.

What makes this work significant is how it changes the workflow for scientists. Previously, extracting these numbers from complex, multi-step reactions was a slow, iterative process that often relied on assumptions about how the reactions were distributed. This new method removes the need for those assumptions. Once the network is trained, it can look at a single set of experimental data and immediately output the kinetic parameters without needing to run further optimizations or guesswork. The researchers found that the method works reliably for both simple and moderately complex systems, providing a fast and direct path from raw experimental data to a deep understanding of the material's behavior. While the current study focused on materials where the reaction steps are distinct, the authors note that future work will need to address even more complex mixtures where reaction peaks overlap heavily. For now, however, this physics-informed approach offers a robust and efficient tool for decoding the thermal stories of complex materials, turning what was once a difficult inverse problem into a straightforward prediction.

Technical Summary: Physics-Informed Deep Neural Operator for Kinetic Parameter Estimation in Parallel Reaction Systems

Problem Statement
Thermogravimetric analysis (TGA) is a critical tool for characterizing the thermal stability and decomposition mechanisms of materials. However, accurately estimating kinetic parameters for complex, multi-step pyrolysis systems remains a significant challenge. Traditional methods (e.g., Freeman–Carroll, Kissinger, Coats–Redfern) often focus on single-step models or require specific prior assumptions about reaction mechanisms. While the Distributed Activation Energy Model (DAEM) offers a flexible framework for describing complex kinetics as a superposition of parallel first-order reactions, solving the associated inverse problem is difficult. Conventional approaches typically rely on iterative optimization to minimize the discrepancy between predicted and experimental conversion curves. These methods are computationally expensive, prone to convergence issues, and often require prescribing the functional form of the activation energy distribution (e.g., Gaussian), which may not reflect the true underlying physics of heterogeneous materials.

Methodology
The authors propose a novel approach using a Physics-Informed Deep Neural Operator (PI-DeepONet) to estimate kinetic parameters for the discrete DAEM without iterative optimization or prior distributional assumptions.

Mathematical Formulation: The study adopts a discrete approximation of the DAEM, modeling the overall conversion $\alpha(t)$ as a weighted sum of $N$ parallel first-order reactions. The kinetic laws are governed by the Arrhenius equation, where the goal is to determine the pre-exponential factors ( $A_i$ ), activation energies ( $E_i$ ), and weights ( $c_i$ ) given experimental conversion data and heating rates.
Operator Learning Framework: Instead of treating parameter estimation as a standard inverse optimization problem, the authors frame it as an operator learning task. They define an operator $G$ that maps the input function space (conversion curve $\alpha(t)$ and heating rate $\beta$ ) directly to the output parameter space ( $A_i, E_i$ ).
Network Architecture: The PI-DeepONet consists of two sub-networks:
- Branch Net: Encodes the input conversion profile $\alpha(t)$ .
- Trunk Net: Encodes the heating rate $\beta$ .
- The outputs are combined to predict the kinetic parameters.
Physics-Informed Loss Function: To ensure physical consistency and reduce the need for large paired datasets, the loss function combines two components:
1. Parameter Prediction Error: The difference between predicted and target kinetic parameters (in the training phase).
2. Physics Reconstruction Error: The network's predicted parameters are fed back into the governing DAEM equations to reconstruct the conversion curve. The loss includes the discrepancy between this reconstructed curve and the original input curve.
Data Strategy: The model is trained on synthetic data generated by numerically solving the forward DAEM problem across a wide range of parameter values. Data preprocessing involves logarithmic transformations to ensure positive outputs for physical parameters.

Key Contributions

Operator Learning for Inverse Problems: The study applies the PI-DeepONet framework to the specific domain of pyrolysis kinetics, treating parameter estimation as a direct mapping from function space to parameter space.
Efficiency and Direct Prediction: Unlike traditional iterative methods, the proposed approach requires only a single training phase. Once trained, it can directly predict kinetic parameters from experimental conversion data in a single forward pass, eliminating the need for repeated optimization.
No Prior Assumptions: The method does not require prescribing the shape of the activation energy distribution (e.g., Gaussian) or assuming constant pre-exponential factors across reactions, allowing for a more flexible representation of complex reaction systems.
Validation: The method is rigorously tested on both synthetic one-step and two-step reaction models and validated against experimental TGA data for calcium oxalate monohydrate.

Results

Synthetic Data Performance:
- One-Step Model: The model achieved extremely high accuracy, with relative $L_2$ errors in conversion reconstruction below $10^{-3}$ and coefficients of determination ( $R^2$ ) exceeding 0.99999 across various heating rates.
- Two-Step Model: The approach successfully estimated parameters for a more complex two-step system. While errors were slightly higher than the one-step case, $R^2$ values remained above 0.999, and the reconstructed conversion and reaction rate curves closely matched target values.
Experimental Validation:
- The model was applied to experimental TGA data of calcium oxalate monohydrate, a material known for its distinct multi-step decomposition (dehydration, CO release, CO2 elimination).
- For the first reaction step, the predicted kinetic parameters yielded conversion and mass loss curves that agreed well with experimental measurements.
- For the first two steps combined, the reconstructed curves also demonstrated close agreement with experimental data, confirming the method's ability to handle overlapping reaction stages in real-world scenarios.

Significance and Claims
The paper claims that the PI-DeepONet approach provides an efficient and reliable solution for the online kinetic analysis of complex multi-step pyrolysis systems. By embedding physical laws directly into the neural operator architecture, the method overcomes the computational bottlenecks and convergence difficulties associated with traditional iterative inverse solvers. The authors emphasize that their method enables direct parameter prediction after a single training procedure, making it suitable for practical engineering applications where rapid analysis of thermogravimetric data is required.

The study acknowledges limitations, noting that the current framework assumes the number of parallel reactions ( $N$ ) is known in advance and that the results are based on relatively clean data. Future work is proposed to address automatic determination of $N$ , robustness to measurement noise, and application to complex biomass components (cellulose, hemicellulose, lignin). However, within the scope of this work, the method is presented as a significant advancement in the speed and reliability of kinetic parameter estimation for distributed activation energy models.

Technical Summary: Physics-Informed Deep Neural Operator for Kinetic Parameter Estimation in Parallel Reaction Systems

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