Franz Martin Rohrhofer

Physics-informed neural networks (PINNs) have emerged as a promising deep learning method, capable of solving forward and inverse problems governed by differential equations. Despite their recent advance, it is widely acknowledged that PINNs are difficult to train and often require a careful tuning of loss weights when data and physics loss functions are combined by scalarization of a multi-objective (MO) problem. In this paper, we aim to understand how parameters of the physical system, such as characteristic length and time scales, the computational domain, and coefficients of differential equations affect MO optimization and the optimal choice of loss weights. Through a theoretical examination of where these system parameters appear in PINN training, we find that they effectively and individually scale the loss residuals, causing imbalances in MO optimization with certain choices of system parameters. The immediate effects of this are reflected in the apparent Pareto front, which we define as the set of loss values achievable with gradient-based training and visualize accordingly. We empirically verify that loss weights can be used successfully to compensate for the scaling of system parameters, and enable the selection of an optimal solution on the apparent Pareto front that aligns well with the physically valid solution. We further demonstrate that by altering the system parameterization, the apparent Pareto front can shift and exhibit locally convex parts, resulting in a wider range of loss weights for which gradient-based training becomes successful. This work explains the effects of system parameters on MO optimization in PINNs, and highlights the utility of proposed loss weighting schemes.

Most engineering domains abound with models derived from first principles that have beenproven to be effective for decades. These models are not only a valuable source of knowledge, but they also form the basis of simulations. The recent trend of digitization has complemented these models with data in all forms and variants, such as process monitoring time series, measured material characteristics, and stored production parameters. Theory-inspired machine learning combines the available models and data, reaping the benefits of established knowledge and the capabilities of modern, data-driven approaches. Compared to purely physics- or purely data-driven models, the models resulting from theory-inspired machine learning are often more accurate and less complex, extrapolate better, or allow faster model training or inference. In this short survey, we introduce and discuss several prominent approaches to theory-inspired machine learning and show how they were applied in the fields of welding, joining, additive manufacturing, and metal forming.