Deep Operator Networks for Surrogate Modeling of Cyclic Adsorption Processes with Varying Initial Conditions

TL;DR

The paper addresses surrogate modeling of cyclic adsorption processes by learning the gas-phase and solid-phase solution operators that map the initial gas concentration profile $C_g^{*0}(\xi^*)$ to the spatiotemporal fields $C_g^*(\xi^*,\tau^*)$ and $C_s^*(\xi^*,\tau^*)$ using DeepONets. A large physically motivated dataset of 10,000 initial conditions drawn from four analytic families is used, with a 6×200-neuron architecture for both branch and trunk, and a loss combining initial-condition consistency $L_{ic}$ and full-field data loss $L_{data}$. The method generalizes well to out-of-distribution initial conditions, achieving an average relative $L^2$ error of $0.1684\%$ on the test set and $2.282\%$ on the expanded OOD set, while errors concentrate near sharp adsorption front regions due to spectral bias. These results suggest DeepONets can serve as efficient surrogates for accelerating cyclic adsorption simulations and optimization, with potential improvements such as hard boundary constraints, Fourier feature mappings, or POD-informed trunks to mitigate localized high-frequency errors.

Abstract

Deep Operator Networks are emerging as fundamental tools among various neural network types to learn mappings between function spaces, and have recently gained attention due to their ability to approximate nonlinear operators. In particular, DeepONets offer a natural formulation for PDE solving, since the solution of a partial differential equation can be interpreted as an operator mapping an initial condition to its corresponding solution field. In this work, we applied DeepONets in the context of process modeling for adsorption technologies, to assess their feasibility as surrogates for cyclic adsorption process simulation and optimization. The goal is to accelerate convergence of cyclic processes such as Temperature-Vacuum Swing Adsorption (TVSA), which require repeated solution of transient PDEs, which are computationally expensive. Since each step of a cyclic adsorption process starts from the final state of the preceding step, effective surrogate modeling requires generalization across a wide range of initial conditions. The governing equations exhibit steep traveling fronts, providing a demanding benchmark for operator learning. To evaluate functional generalization under these conditions, we construct a mixed training dataset composed of heterogeneous initial conditions and train DeepONets to approximate the corresponding solution operators. The trained models are then tested on initial conditions outside the parameter ranges used during training, as well as on completely unseen functional forms. The results demonstrate accurate predictions both within and beyond the training distribution, highlighting DeepONets as potential efficient surrogates for accelerating cyclic adsorption simulations and optimization workflows.

Figures (11)

• Figure 1: Illustration of the input functions dataset, where $C^*$ represents either $C_g^*$ or $C_s^*$. Each title reports the functional type and the number of its position in the list of functions.
• Figure 2: Illustration of the DeepONet in the problem setup, in which the spatio-temporal coordinates as well as the input functions are fed to the trunk and the branch net, respectively. The network provides solutions in the spatio-temporal domain.
• Figure 3: Training and validation losses trends per epoch for the gas phase DeepONet.
• Figure 4: Parity plot of $C_g^*$ for the test set, with an average $L^2$ relative error of $0.1684\%$.
• Figure 5: Heatmaps (\ref{['fig:heatmaps309']}) of the prediction and the absolute point-wise error, and time snapshots (\ref{['fig:snapshots309']}) of the true and predicted results for a test example with a sigmoidal initial condition function.