Wrik Mallik

Wrik Mallik, Rajeev K. Jaiman, Jasmin Jelovica

It is challenging to construct generalized physical models of wave propagation in nature owing to their complex physics as well as widely varying environmental parameters and dynamical scales. In this article, we present the convolutional autoencoder recurrent network (CRAN) as a data-driven model for learning wave propagation phenomena. The CRAN consists of a convolutional autoencoder for learning low-dimensional system representation and a long short-term memory recurrent neural network for the system evolution in low dimension. We show that the convolutional autoencoder significantly outperforms the dimension-reduction of complex wave propagation phenomena via projection-based methods as it can directly learn subspaces resembling wave characteristics. On the other hand, the projection-based modes are restricted to the Fourier subspace. Geometric priors of the convolutional autoencoder enabling selective scale separation of complex wave dynamics further enhance its dimension-reduction capability. We also demonstrate that geometric priors such as translation equivariance and translational invariance of the convolutional autoencoder enable generalized learning of low-dimensional maps. Thus, the composite CRAN model connecting the convolutional autoencoder with a long short-term memory network specially designed for autoregressive modeling can perform generalized wave propagation prediction over the desired time horizon. Numerical experiments display 90% mean structural similarity index measure of CRAN predictions compared to true solutions for out-of-training cases, and less than 10% pointwise $L_1$ error for most cases, verifying such generalization claims. Finally, the CRAN predictions offer similar wave characteristic patterns to the target solutions indicating not only their generalization but also their kinematical consistency.

Wrik Mallik, Neil Farvolden, Jasmin Jelovica et al.

This article presents a reduced-order modeling methodology via deep convolutional neural networks (CNNs) for shape optimization applications. The CNN provides a nonlinear mapping between the shapes and their associated attributes while conserving the equivariance of these attributes to the shape translations. To implicitly represent complex shapes via a CNN-applicable Cartesian structured grid, a level-set method is employed. The CNN-based reduced-order model (ROM) is constructed in a completely data-driven manner thus well suited for non-intrusive applications. We demonstrate our ROM-based shape optimization framework on a gradient-based three-dimensional shape optimization problem to minimize the induced drag of a wing in low-fidelity potential flow. We show a good agreement between ROM-based optimal aerodynamic coefficients and their counterparts obtained via a potential flow solver. The predicted behavior of the optimized shape is consistent with theoretical predictions. We also present the learning mechanism of the deep CNN model in a physically interpretable manner. The CNN-ROM-based shape optimization algorithm exhibits significant computational efficiency compared to the full-order model-based online optimization applications. The proposed algorithm promises to develop a tractable DL-ROM-driven framework for shape optimization and adaptive morphing structures.

Wrik Mallik, Rajeev K. Jaiman, Jasmin Jelovica

Although machine learning (ML) is increasingly employed recently for mechanistic problems, the black-box nature of conventional ML architectures lacks the physical knowledge to infer unforeseen input conditions. This implies both severe overfitting during a dearth of training data and inadequate physical interpretability, which motivates us to propose a new kinematically consistent, physics-based ML model. In particular, we attempt to perform physically interpretable learning of inverse problems in wave propagation without suffering overfitting restrictions. Towards this goal, we employ long short-term memory (LSTM) networks endowed with a physical, hyperparameter-driven regularizer, performing penalty-based enforcement of the characteristic geometries. Since these characteristics are the kinematical invariances of wave propagation phenomena, maintaining their structure provides kinematical consistency to the network. Even with modest training data, the kinematically consistent network can reduce the $L_1$ and $L_\infty$ error norms of the plain LSTM predictions by about 45% and 55%, respectively. It can also increase the horizon of the plain LSTM's forecasting by almost two times. To achieve this, an optimal range of the physical hyperparameter, analogous to an artificial bulk modulus, has been established through numerical experiments. The efficacy of the proposed method in alleviating overfitting, and the physical interpretability of the learning mechanism, are also discussed. Such an application of kinematically consistent LSTM networks for wave propagation learning is presented here for the first time.