Multi-scale Time-stepping of Partial Differential Equations with Transformers

Developing fast surrogates for Partial Differential Equations (PDEs) will accelerate design and optimization in almost all scientific and engineering applications. Neural networks have been receiving ever-increasing attention and demonstrated remarkable success in computational modeling of PDEs, however; their prediction accuracy is not at the level of full deployment. In this work, we utilize the transformer architecture, the backbone of numerous state-of-the-art AI models, to learn the dynamics of physical systems as the mixing of spatial patterns learned by a convolutional autoencoder. Moreover, we incorporate the idea of multi-scale hierarchical time-stepping to increase the prediction speed and decrease accumulated error over time. Our model achieves similar or better results in predicting the time-evolution of Navier-Stokes equations compared to the powerful Fourier Neural Operator (FNO) and two transformer-based neural operators OFormer and Galerkin Transformer.

Surrogate Modeling of Melt Pool Thermal Field using Deep Learning

Powder-based additive manufacturing has transformed the manufacturing industry over the last decade. In Laser Powder Bed Fusion, a specific part is built in an iterative manner in which two-dimensional cross-sections are formed on top of each other by melting and fusing the proper areas of the powder bed. In this process, the behavior of the melt pool and its thermal field has a very important role in predicting the quality of the manufactured part and its possible defects. However, the simulation of such a complex phenomenon is usually very time-consuming and requires huge computational resources. Flow-3D is one of the software packages capable of executing such simulations using iterative numerical solvers. In this work, we create three datasets of single-trail processes using Flow-3D and use them to train a convolutional neural network capable of predicting the behavior of the three-dimensional thermal field of the melt pool solely by taking three parameters as input: laser power, laser velocity, and time step. The CNN achieves a relative Root Mean Squared Error of 2% to 3% for the temperature field and an average Intersection over Union score of 80% to 90% in predicting the melt pool area. Moreover, since time is included as one of the inputs of the model, the thermal field can be instantly obtained for any arbitrary time step without the need to iterate and compute all the steps