Speeding Up Reactive Transport Simulations in Cement Systems by Surrogate Geochemical Modeling: Deep Neural Networks and k-Nearest Neighbors

Abstract

We accelerate reactive transport (RT) simulation by replacing the geochemical solver in the RT code by a surrogate model or emulator, considering either a trained deep neural network (DNN) or a k-nearest neighbor (kNN) regressor. We focus on 2D leaching of hardened cement paste under diffusive or advective-dispersive transport conditions, a solid solution representation of the calcium silicate hydrates and either 4 or 7 chemical components, and use the HPx (coupled Hydrus-PHREEQC model) reactive transport code as baseline. We find that after training, both our DNN-based and kNN-based codes, $\hbox {HPx}_{\rm{py}}$-DNN and $\hbox {HPx}_{\rm{py}}$-kNN, can make satisfactorily to very accurate predictions while providing either a 3 to 9 speedup factor compared to HPx with parallelized geochemical calculations over 4 cores. Benchmarking against single-threaded HPx, these speedup factors become 8 to 33. Overall, $\hbox {HPx}_{\rm{py}}$-DNN and $\hbox {HPx}_{\rm{py}}$-kNN are found to achieve a close to optimal speedup when DNN regression and kNN search are performed on a GPU. Importantly, for the more complex 7-components cement system, no emulator that is globally accurate over the full space of possible geochemical conditions could be devised. Instead we therefore build “local” emulators that are only valid over a relevant fraction of the input parameter space. This space is identified by running a coarse and thus computationally cheap full RT simulation, and subsequently explored by kernel density sampling. Future work will focus on improving accuracy for this type of cement systems.

Appendix: Details of the Used DNN

Let us denote the input and output space dimensions by $N_{\rm{in}}$ and $N_{\rm{out}}$, and a given reference number of hidden neurons as $NN_{\rm{h}}$. A fully connected layer (FC) with a selu activation function taking a $N_{\rm{in}}$-dimensional input and producing a $N_{\rm{out}}$-dimensional output using a self-exponential activation function (SELU) is then referred to as $\left[ FC_{\rm{SELU}}-I_{N_{\rm{in}}}-O_{N_{\rm{out}}}\right]$. We call $\left[ FC_{\rm{LIN}}-I_{N_{\rm{in}}}-O_{N_{\rm{out}}}\right]$ the same layer but with a linear activation function. From input to output layer, our DNNs are built as follows

• $\left[ FC_{\rm{SELU}}-I_{N_{\rm{in}}}-O_{N_{\rm{h}}}\right]$

• $\left[ FC_{\rm{SELU}}-I_{N_{\rm{h}}}-O_{2N_{\rm{h}}}\right]$

• $\left[ FC_{\rm{SELU}}-I_{2N_{\rm{h}}}-O_{4N_{\rm{h}}}\right]$

• $\left[ FC_{\rm{SELU}}-I_{4N_{\rm{h}}}-O_{2N_{\rm{h}}}\right]$

• $\left[ FC_{\rm{SELU}}-I_{2N_{\rm{h}}}-O_{N_{\rm{h}}}\right]$

• $\left[ FC_{\rm{LIN}}-I_{N_{\rm{h}}}-O_{N_{\rm{out}}}\right]$

We set $N_{\rm{h}} = 32$ for cement system 1 ($N_{\rm{in}} = 2$, $N_{\rm{out}} = 4$) and $N_{\rm{h}} = 128$ for cement system 2 ($N_{\rm{in}} = 5$, $N_{\rm{out}} = 7$).