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With pleasure we announce the online seminar series on Scientific Learning organized by the PhD school FoMICS-DADSi at USI Lugano/Switzerland.<br>
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The first seminar will be given on 19 May at 5:00 pm (UTC+2:00 Zurich time) by Prof. Paris Perdikaris (Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania).<br>
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The seminar will consist of a lecture plus a scientific talk.<br>
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Topic of the lecture: Making neural networks physics-informed<br>
Topic of the scientific talk: Learning the solution operator of parametric partial differential equations with physics-informed DeepONets.<br>
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The event will be virtual. To register, please contact <a href="mailto:nestom@usi.ch" target="_blank">nestom@usi.ch</a><mailto:<a href="mailto:nestom@usi.ch" target="_blank">nestom@usi.ch</a>>.<br>
Further details can be found at <a href="https://fomics.usi.ch/index.php/workshops/15-ics/306-pinn" rel="noreferrer" target="_blank">https://fomics.usi.ch/index.php/workshops/15-ics/306-pinn</a><br>
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Part A: Lecture<br>
Making neural networks physics-informed<br>
Leveraging advances in automatic differentiation, physics-informed neural networks are introducing a new paradigm in tackling forward and inverse problems in computational mechanics. Under this emerging paradigm, unknown quantities of interest are typically parametrized by deep neural networks, and a multi-task learning problem is posed with the dual goal of fitting observational data and approximately satisfying a given physical law, mathematically expressed via systems of partial differential equations (PDEs). PINNs have demonstrated remarkable flexibility across diverse applications, but, despite some empirical success, a concrete mathematical understanding of the mechanisms that render such constrained neural network models effective is still lacking. In fact, more often than not, PINNs are notoriously hard to train, especially for forward problems exhibiting high-frequency or multi-scale behavior. In this talk we will discuss the basic principles of making neural networks physics informed with an emphasis on the caveats one should be aware of and how those can be addressed in practice.<br>
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Part B: Seminar<br>
Learning the solution operator of parametric partial differential equations with physics-informed DeepONets<br>
Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise, they typically require large training data-sets consisting of paired input-output observations which may be expensive to obtain, while their predictions may not be consistent with the underlying physical principles that generated the observed data. In this work, we propose a novel model class coined as physics-informed DeepONets, which introduces an effective regularization mechanism for biasing the outputs of DeepOnet models towards ensuring physical consistency. This is accomplished by leveraging automatic differentiation to impose the underlying physical laws via soft penalty constraints during model training. We demonstrate that this simple, yet remarkably effective extension can not only yield a significant improvement in the predictive accuracy of DeepOnets, but also greatly reduce the need for large training data-sets. To this end, a remarkable observation is that physics-informed DeepONets are capable of solving parametric partial differential equations (PDEs) without any paired input-output observations, except for a set of given initial or boundary conditions. We illustrate the effectiveness of the proposed framework through a series of comprehensive numerical studies across various types of PDEs. Strikingly, a trained physics informed DeepOnet model can predict the solution of O(1000) time-dependent PDEs in a fraction of a second -- up to three orders of magnitude faster compared to a conventional PDE solver.<br>
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Prof. Dr. Rolf Krause<br>
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Chair for Advanced Scientific Computing<br>
Group High Performance Methods for Numerical Simulation in Science, Medicine and Engineering<br>
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Euler Institute<br>
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Center for Computational Medicine in Cardiology<br>
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<a href="https://www.euler.usi.ch" rel="noreferrer" target="_blank">https://www.euler.usi.ch</a><<a href="https://www.euler.usi.ch/" rel="noreferrer" target="_blank">https://www.euler.usi.ch/</a>><br>
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Università della Svizzera italiana<br>
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East Campus - Sector D<br>
Office D5.15<br>
Via la Santa 1<br>
6962 Lugano-Viganello, Switzerland<br></div></div>