In this article, the authors adapt and utilize data-driven advanced image processing and machine learning techniques to identify the underlying dynamics and the model parameters for dynamic processes driven by partial differential equations (PDEs). Potential applications include non-destructive inspection for material crack detection using thermal imaging as well as real-time anomaly detection for process monitoring of three-dimensional printing applications. A neural network (NN) architecture is established that offers sufficient flexibility for spatial and temporal derivatives to capture the physical dependencies inherent in the process. Predictive capabilities are then established by propagating the process forward in time using the acquired model structure as well as individual parameter values. Moreover, deviations in the predicted values can be monitored in real time to detect potential process anomalies or perturbations. For concept development and validation, this article utilizes well-understood PDEs such as the homogeneous heat diffusion equation. Time series data governed by the heat equation representing a parabolic PDE is generated using high-fidelity simulations in order to construct the heat profile. Model structure and parameter identification are realized through a shallow residual convolutional NN. The learned model structure and associated parameters resemble a spatial convolution filter, which can be applied to the current heat profile to predict the diffusion behavior forward in time.