Conference Agenda

By now, simulations have become an essential tool in engineering sciences. Especially in the field of production engineering, many production processes do not easily allow for measurements. Thus, digital twins, e.g., in the form of high-fidelity simulation models, gain increasing interest as they enable insights into the underlying dynamics of the manufacturing process. However, simulating realistic applications with conventional full-order models is often very expensive due to the large number of degrees of freedom.

This motivates the interest in model-order reduction techniques, which approximate full-order models at much lower computational costs by drastically reducing the degrees of freedom. Here, the recent breakthroughs in deep-learning approaches have drawn attention toward data-based strategies for constructing reduced-order models as alternatives to the well-established full-order models. Many deep-learning-based approaches rely on the abundance of data, which is usually scarce in engineering applications. Utilizing the underlying physics additionally for guiding the learning process has become particularly attractive for constructing accurate but fast digital twins.

In our work, we are interested in the plastic manufacturing process of profile extrusion. Precisely, we are interested in modeling the shear-thinning flow of the highly viscous plastics melt inside profile extrusion dies. To construct a digital twin, we utilize Physics-Informed Neural Networks [1]. We will present comparisons with respect to high-fidelity digital twins, i.e., provided through Finite Element simulation results, and elaborate on training heuristics, which proved essential for our application to obtain reduced models with sufficient accuracy.

References

[1] Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707. https://doi.org/10.1016/j.jcp.2018.10.045

Physics-Informed Neural Networks (PINNs) have been introduced as a promising method that can combine differential equations and measurement data in the loss function of the neural network [1]. PINNs are a meshless method so they can handle high-dimensional domains. Furthermore, they are a good candidate to solve inverse problems due to the easy integration of data. Based on sensor data, PINNs can be used as a surrogate fast-to-evaluate model in hybrid digital twins of civil engineering structures [2].

One of the main challenges is to find a suitable PINN configuration since the prediction accuracy and model efficiency depend on hyperparameters [3]. Commonly, hyperparameters have been determined by manual adjustment through trial and error. In this contribution, we investigate the network and optimizer parameters of PINNs in various examples aiming for a hyperparameter tuning guideline for computational mechanics problems.

The search space for the network hyperparameters contains the distribution of training points, the number of hidden layers with accompanying neurons, activation functions, and network parameter initializers. On the other hand, the investigated optimizer parameters consist of different optimization algorithms along with their combinations, learning rates and the number of iterations. The main targets of hyperparameter optimization are training performance, loss on collocation and boundary points, and prediction accuracy. Besides a systematic exploration of the search space, we attempt a sensitivity analysis of the optimal PINN configuration in dependence on varying material parameters.

Specific examples include a one-dimensional cantilever beam under a triangular distributed load, a two-dimensional Lam‘e problem and a Hertzian contact problem with the mixed-variable formulation, as well as two- and three-dimensional heat transfer problems and corresponding inverse problems.

[1] M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational Physics, vol. 378, pp. 686–707, 2019.

[2] M. von Danwitz, T. T. Kochmann, T. Sahin, J. Wimmer, T. Braml, and A. Popp, “Hybrid digital twins: A proof of concept for reinforced concrete beams,” Accepted in Proceedings in Applied Mathematics and Mechanics, 2022.

[3] Y. Wang, X. Han, C.-Y. Chang, D. Zha, U. Braga-Neto, and X. Hu, “Auto-pinn: Understanding and

optimizing physics-informed neural architecture,” arXiv preprint arXiv:2205.13748, 2022.

Stirred Tank Reactors (STRs) play a central role in biotechnological process development and manufacturing. Digital twins of STRs can be used both to minimize the amount of supporting experimental studies required during process design and scale-up, and to deepen the understanding of conditions inside a reactor, where little information is available due to the lack of appropriate measurement techniques. For this purpose, Computational Fluid Dynamics (CFD) tools are already widely used in the industry. However, the high computational cost of high-fidelity simulations, especially in scenarios, where the same model must be solved repeatedly for different parameter values (such as stirring rate), motivates the construction of less computationally intensive Reduced Order Models (ROMs) to approximate solutions. Physics-Informed Neural Networks (PINNs), originally proposed by Raissi et al. [1], are a promising candidate for ROMs in engineering problems, as they allow to simultaneously exploit both the available data and the knowledge of the underlying physics of the problem by embedding the governing equations in the loss function of the neural network.

This use case represents a particular challenge for PINNs due to the geometric complexity of the computational domain and the large variety of phenomena involved in the process (e.g., turbulence, mass transfer).

Building on the investigation of strategies to improve the predictive accuracy of the model, for example by imposing boundary constraints in a post-processing step using an interpolation spline as proposed in [2] or by leveraging additional knowledge of the problem, such as domain decomposition based on the different character of the flow in different parts of the domain, we aim to apply the approaches tested in 2D to more realistic 3D models. The presented methods can be transfered to other complex problems to improve the overall performance of PINNs.

References

[1] M. Raissi, P. Perdikaris, and G. E. Karniadakis, \Physics-informed neural networks: A deep

learning framework for solving forward and inverse problems involving nonlinear partial di er-

ential equations," Journal of Computational Physics, vol. 378, pp. 686{707, Feb. 2019.

[2] H. Sheng and C. Yang, \Pfnn: A penalty-free neural network method for solving a class of second-

order boundary-value problems on complex geometries," Journal of Computational Physics,

vol. 428, p. 110085, 2021.