Physics-Guided Conditional Diffusion Model for Rare Event Synthesis and Diagnosis for the Water-Gas Shift Reaction

Md Abrar Rafid Siddique; Bibek Aryal; Qiugang Lu

cs.LGSep 28, 2026

Physics-Guided Conditional Diffusion Model for Rare Event Synthesis and Diagnosis for the Water-Gas Shift Reaction

Authors: Md Abrar Rafid Siddique, Bibek Aryal, Qiugang Lu

Organizations: Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA

Abstract

As the world moves towards sustainable energy sources, hydrogen (H2) can be treated as an eco-friendly alternative to fossil fuels due to its high energy density and zero carbon emissions. The water-gas shift (WGS) reaction is a widely used industrial process for hydrogen production by converting carbon monoxide and steam into hydrogen and carbon dioxide. However, occurrences like severe fouling, catalyst deterioration, and thermal runaway can hamper the reaction kinetics/process safety and decrease the yield of H2. These incidents are rare, and gathering process data under such abnormal conditions is challenging. In this work, we propose a physics-guided conditional diffusion model to generate realistic rare-event trajectories for the WGS reaction. The proposed model integrates a conditional denoising diffusion probabilistic model (CDDPM) with governing laws of the reaction to generate physically consistent process trajectories. The conditioning features allow the model to produce high-quality synthetic profiles for rare-event domains that are typically beyond the training regimes. The generated rare-event trajectories then augment the raw dataset for a balanced distribution between normal and abnormal conditions. We further propose a hazard score to assess the risk severity of the operating condition based on the operating trajectory. Deep learning models are trained with the augmented dataset to diagnose the health status of the reaction. Simulation results show that the proposed physics-guided diffusion model outperforms data-driven models in terms of the quality of synthetic data and diagnosis performance for rare events.

Figures & tables

Figure 1 : The overall framework of the proposed Pg-CDDPM-based rare-event synthesis and diagnosis method.

Figure 2 : Overview of the forward and reverse processes of a diffusion model.

Figure 3 : Schematic diagram of the WGS reaction in a continuous stirred-tank reactor.

Figure 4 : Overview of the proposed Pg-CDDPM framework.

Figure 5 : Architecture of the proposed conditional 1D U-Net used to predict the added noise during the reverse diffusion process.

Figure 6 : Architecture of the ResBlock 1D used as building blocks of the diffusion model.

Layer	Operation	Kernel	Stride	Channels (In $\rightarrow$ Out)	Trajectory (In $\rightarrow$ Out)
Input	Concatenate inputs	–	–	$7+2\rightarrow 9$	$L\rightarrow L$
Resnet Block 1	ResBlock 1D	3	1	$9\rightarrow\mathrm{base\_ch}$	$L\rightarrow L$
Maxpooling 1	MaxPool1D	2	2	$\mathrm{base\_ch}\rightarrow\mathrm{base\_ch}$	$L\rightarrow\frac{L}{2}$
Resnet Block 2	ResBlock 1D	3	1	$\mathrm{base\_ch}\rightarrow 2\times\mathrm{base\_ch}$	$\frac{L}{2}\rightarrow\frac{L}{2}$
Maxpooling 2	MaxPool1D	2	2	$2\times\mathrm{base\_ch}\rightarrow 2\times\mathrm{base\_ch}$	$\frac{L}{2}\rightarrow\frac{L}{4}$
Bottleneck	ResBlock 1D	3	1	$2\times\mathrm{base\_ch}\rightarrow 4\times\mathrm{base\_ch}$	$\frac{L}{4}\rightarrow\frac{L}{4}$

Table 1 : Summary of the proposed 1D U-Net architecture.

Figure 7 : The flow chart for training the proposed Pg-CDDPM to generate high-fidelity synthetic profiles for the WGS reaction.

Figure 8 : Hazard-based classification of generated reactor trajectories, classification thresholds, and temperature- and hydrogen-related contributions to the hazard score.

Parameter	Value	Parameter	Value
Deep-learning framework	PyTorch	Beta schedule	Linear
Optimizer	Adam	$\beta_{\mathrm{start}}$	$10^{-4}$
Learning rate	$10^{-4}$	$\beta_{\mathrm{end}}$	$10^{-2}$
Loss function	MSE	Input channels	9
Batch size	5	Output channels	7
Number of epochs	6000	Base channels	64

Table 2 : Hyperparameters used for training the proposed Pg-CDDPM.

Parameter	Value	Parameter	Value
Deep-learning framework	PyTorch	Fully connected activation	ReLU
Sequence length	100 time steps	Number of output classes	4
Number of input features	7	Loss function	Cross-entropy
Input features	CO, H 2 O, CO 2 , H 2 , $T$ , $T_{c}$ , $q$	Optimizer	Adam
GRU hidden size	64	Learning rate	$10^{-3}$
Number of GRU layers	8	Batch size	50

Table 3 : Training hyperparameters and architecture of the GRU classifier.

Input Variable	Symbol	Range
Coolant temperature	$T_{c}$	$350$ – $500~\mathrm{K}$
Feed flow rate	$F$	$70$ – $150~\mathrm{L\,min^{-1}}$
Input holding time	$N_{\mathrm{hold}}$	$10$ time steps

Table 4: Ranges of input variables used for generating the input profiles.

Initial reactor conditions			Feed conditions
Parameter	Symbol	Value	Parameter	Symbol	Value
Initial CO concentration	$C_{\mathrm{CO},0}$	$1.5$ – $2.0$	Feed CO concentration	$C_{f,\mathrm{CO}}$	$1.5$
Initial H 2 O concentration	$C_{\mathrm{H_{2}O},0}$	$3.0$ – $3.5$	Feed H 2 O concentration	$C_{f,\mathrm{H_{2}O}}$	$3.0$
Initial CO 2 concentration	$C_{\mathrm{CO_{2}},0}$	$0.2$	Feed CO 2 concentration	$C_{f,\mathrm{CO_{2}}}$	$0.0$
Initial H 2 concentration	$C_{\mathrm{H_{2}},0}$	$0.2$	Feed H 2 concentration	$C_{f,\mathrm{H_{2}}}$	$0.0$
Initial reactor temperature	$T_{0}$	$500$ – $550$	Feed temperature	$T_{f}$	$550$

Table 5 : Initial reactor and feed conditions used for generating the training trajectories. Concentrations are in

\mathrm{mol\,L^{-1}}

and temperatures are in

\mathrm{K}

Parameter	Symbol	Value	Parameter	Symbol	Value
Reactor volume	$V$	$100~\mathrm{L}$	Base overall heat transfer coefficient	$UA$	$5.0\cdot 10^{4}$
Fluid density	$\rho$	$1000~\mathrm{kg\,m^{-3}}$	Heat transfer efficiency factor	$f$	$0.6$ – $1.0$
Heat capacity	$C_{p}$	$0.239~\mathrm{kJ\,kg^{-1}\,K^{-1}}$	Effective heat transfer coefficient	$UA_{\mathrm{eff}}$	$UA\cdot f$
Pre-exponential factor	$k_{0}$	$10^{10}$	Base activation energy	$E_{a}$	$7.2\cdot 10^{4}~\mathrm{J\,mol^{-1}}$
Universal gas constant	$R$	$8.314~\mathrm{J\,mol^{-1}\,K^{-1}}$	Activation energy factor	$f_{E_{a}}$	$1.0$ – $1.5$
Heat of reaction	$\Delta H_{r}$	$-4.1\cdot 10^{4}~\mathrm{J\,mol^{-1}}$	Effective activation energy	$E_{a,\mathrm{eff}}$	$E_{a}\cdot f_{E_{a}}$

Table 6 : Process parameters used in the WGS reactor model.

Trajectory No.	Heat Transfer Efficiency Factor $f$	Activation Energy Factor $f_{E_{a}}$
1–30	0.6–1.0	1.0–1.2
31–60	0.6–1.0	1.2–1.3
61–90	0.6–1.0	1.3–1.4
91–120	0.6–1.0	1.4–1.45
121–130	0.6–1.0	1.45–1.5

Table 7: Ranges of degradation factors used to generate the training trajectories.

Figure 9 : Comparison between synthetic and ground-truth reaction profiles (interpolation) for heat transfer efficiency $f=0.72$ and activation energy $f_{E_{a}}=1.30$ without the physics-guided loss.

Figure 10 : Comparison between synthetic and ground-truth reaction profiles (interpolation) for heat transfer efficiency $f=0.72$ and activation energy $f_{E_{a}}=1.30$ with the proposed Pg-CDDPM.

Model	Temperature (K)	CO	H 2 O	H 2
Without physics-guided loss	$5.0$	0.0368	0.0377	0.0348
With physics-guided loss	4.0	0.0283	0.0264	0.0267

Table 8: RMSE comparison for the diffusion model without and with physics-guided loss for the interpolation problem at

f=0.72

and

f_{E_{a}}=1.30

Figure 11 : Comparison between synthetic and ground-truth reaction profiles (extrapolation) for heat transfer efficiency $f=0.55$ and activation energy $f_{E_{a}}=1.55$ without the physics-guided loss.

Figure 12 : Comparison between synthetic and ground-truth reaction profiles (extrapolation) for heat transfer efficiency $f=0.55$ and activation energy $f_{E_{a}}=1.55$ with the proposed Pg-CDDPM.

Model	Temperature (K)	CO	H 2 O	H 2
Without physics-guided loss	26.6	0.2643	0.2763	0.2573
With physics-guided loss	3.67	0.0262	0.0297	0.0227

Table 9: RMSE comparison without and with the physics-guided loss in the extrapolation region at

f=0.55

and

f_{E_{a}}=1.55

Figure 13 : Comparison of the trajectory temperature RMSE distributions for the conventional C-DDPM and the proposed Pg-CDDPM under interpolation and extrapolation operating conditions.

Figure 14 : Comparison of the trajectory RMSE distributions for the species concentrations under (a) interpolation and (b) extrapolation operating conditions.

Figure 15 : Clustering of the simulator-generated trajectories based on the hazard score with respect to $f$ , $f_{E_{a}}$ , and $H_{2,\mathrm{avg}}$ .

Figure 16 : Distribution of the augmented trajectories based on the hazard score with respect to the $f$ , $f_{E_{a}}$ , and $H_{2,\mathrm{avg}}$ .

Training dataset	Normal	Slight	Extreme	Disaster	Total
Case 1: Simulation	415	415	415	0	1245
Case 2: Simulation + Pg-CDDPM	415	415	415	415	1660
Case 3: Simulation + Conventional CDDPM	415	415	415	415	1660

Table 10: Class distribution of the training datasets for the three cases.

Figure 17 : Confusion matrices of the GRU classifier for the three training cases.

Figure 18 : Comparison of the class-wise precision, recall, and F1-score obtained using the three training datasets.

Metric	Case	Normal	Slightly Abnormal	Extreme	Disaster
	Case 1	0.93	0.95	0.48	0.00
Precision	Case 2	0.93	0.92	0.91	0.94
	Case 3	0.89	0.93	0.51	0.67
	Case 1	0.94	0.92	1.00	0.00
Recall	Case 2	0.93	0.93	0.91	0.92
	Case 3	0.93	0.91	0.88	0.21

Table 11: Comparison of the classification performance for the three training cases. Case 1: Simulation; Case 2: Simulation + Pg-CDDPM; Case 3: Simulation + C-DDPM.

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Industrial time-series signals, such as turbine temperature and rotational speed in aero-engines, are essential for monitoring the health and operational status of complex dynamical systems. However, collecting such data is often limited by harsh environments (e.g., high temperature and high pressure) and the high cost of experimental testing. To address this challenge, we introduce PhysDGM, a stepwise physics-embedded diffusion generative model for synthesizing time-series data that are consistent with the underlying physical laws of dynamical systems. PhysDGM embeds physical laws directly into each reverse diffusion step of the generative process, ensuring trajectory-level physical consistency, rather than enforcing constraints only at the final output. A large-scale AI-synthetic dataset (4.4 million samples, 20x scale-up) constructed by PhysDGM demonstrates strong fidelity across 34 datasets spanning turbofan engines, aero-engines, batteries, and chemical processes. After incorporating the synthetic data, the downstream task performance substantially surpassed that using real data alone by 48% for remaining useful life prediction, 15% for health indicator estimation, 22% for state-of-health assessment, and 20% for fault diagnosis. Moreover, it requires 10-20x less training data than existing approaches, substantially reducing the high cost of data collection in dynamical systems. We further demonstrate PhysDGM's potential in identifying early-stage faults in aero-engines by incorporating AI-synthesized data. In summary, PhysDGM provides a solid foundation for generating physically consistent industrial time-series, paving the way for expanding physics-guided AI into diverse data-scarce environments, including both industrial machinery and complex chemical reaction dynamics.

Abstract

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