Time mesh independent framework for learning materials constitutive relationships

Real-world datasets are rarely populated by evenly distributed entries; unevenness may be caused by sensor malfunctions or randomized sampling due to the process nature. Modeling the constitutive relationship (CR) of materials in scenarios where the temporal data available are uneven is a serious challenge for black box approaches such as artificial neural networks. This work presents a general framework capable of modeling uneven sampled data, which is composed of an Encoder–Decoder (ED) structure. In our framework, the Encoder can process an uneven input sequence, thanks to an approximation of the Ordinary Differential Equations (ODE), and project it into a lower dimensional latent space; the Decoder, on the other hand, can map the compressed information into the output of interest, the material stress response in this work. In the proposed temporal mesh independent framework, the Encoder is a multi-layer structure, with each layer consisting of a Long-Short Term Memory (LSTM) layer, a Closed form Continuous Time (CfC) layer, and a Self Multi-Head Attention Layer (MHAL) layer connected in series. The Decoder can be one Fully Connected Network (FCN) or two FCNs in parallel; in the latter case, the Decoder is capable of giving the mean and the variance of the output. The presented mesh-independent framework demonstrates good accuracy despite both the unevenness and the noise of the training data, specially when its results are compared to the standard ones; thus extending the applicability of neural-network-based black box models in real world applications.