Spatiotemporal Transformation-Based Neural Network With Interpretable Structure for Modeling Distributed Parameter Systems

Many industrial processes can be described by distributed parameter systems (DPSs) governed by partial differential equations (PDEs). In this research, a spatiotemporal network is proposed for DPS modeling without any process knowledge. Since traditional linear modeling methods may not work well for nonlinear DPSs, the proposed method considers the nonlinear space–time separation, which is transformed into a Lagrange dual optimization problem under the orthogonal constraint. The optimization problem can be solved by the proposed neural network with good structural interpretability. The spatial construction method is employed to derive the continuous spatial basis functions (SBFs) based on the discrete spatial features. The nonlinear temporal model is derived by the Gaussian process regression (GPR). Benefiting from spatial construction and GPR, the proposed method enables spatially continuous modeling and provides a reliable output range under the given confidence level. Experiments on a catalytic reaction process and a battery thermal process demonstrate the effectiveness and superiority of the proposed method.