Speaker
Description
Reconstructing nonlinear parametric functions based on sparse measurements or partially observed sensor data is an important problem arising in a number of practical applications such as climate science, fluid dynamics, and biology. To achieve accurate approximation, two crucial aspects need to be considered: (a) choosing the optimal location for data sampling and (b) identifying a suitable model to map the measured data to the corresponding high-dimensional state. Existing approaches to address this challenge either adopt a linear model for reconstruction [1], leading to poor approximations or follow an ad-hoc (random) choice of sensor locations that fail to efficiently recover the true underlying field [2, 3]. Our solution to address this problem uses a neural network architecture that combines the functionalities of data sampling and reconstruction in the form of a pseudo-autoencoder setup. Importantly, it is differentiable and can be trained end-to-end. Once trained, our setup requires only the sparsely sensed data as input, which is then lifted to the true dimension via the decoder. To perform differentiable sampling, we use stochastic optimization, taking advantage of recent developments in statistical techniques to sample from categorical distributions. We illustrate the benefits of the new approach on numerical examples arising in fluid dynamics and climate science.
There is a close similarity between the above framework and hyperreduction methods [4]. We discuss this connection and explore how our method can be put in the same mathematical setting as hyperreduction. This similarity will also illustrate how the new approach is a natural extension of hyperreduction using neural networks.
[1] K. Manohar, B. W. Brunton, J. N. Kutz and S. L. Brunton, Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns, IEEE Control Systems Magazine, vol. 38, no. 3 (2018), pp. 63-86.
[2] N. B. Erichson , L. Mathelin, Z. Yao, S. L. Brunton, M. W. Mahoney, J. N. Kutz, Shallow neural networks for fluid flow
reconstruction with limited sensors, Proceeding of the Royal Society A, 476: 20200097, 2020.
[3] K. Fukami, R. Maulik, N. Ramachandra, K. Fukagata, and K. Taira, Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning, Nature Machine Intelligence 3, no. 11 (2021): 945-951.
[4] C. Farhat, S. Grimberg, A. Manzoni, and A. Quarteroni, Computational bottlenecks for PROMs: precomputation and hyperreduction. In P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W. Schilders & L. Silveira (Ed.), Volume 2 Snapshot-Based Methods and Algorithms, pp. 181-244, De Gruyter, 2021.
