Projects/PINN Operator learning

Phisics-informed Neural Networks for Operator Learning

Many machine learning tasks require working with operators over multidimensional vectors. For example, when working on images, the task is to find an operator that transforms the discretized image into another vector: in the case of a classification task, a probabilistic vector, and in the case of generation, it will also be the image space. If the dimensionality of the space is large, it makes sense to represent the image as a function on the coordinates, yielding a color from the RGB space. A similar problem exists for physical systems. Unlike images, they operate in three-dimensional space. And it is impossible to discretize the space for them. In a general way, this problem is solved by Operator Learning. In this problem, it is proposed to learn an operator on a continuous space. It is proposed to extend this idea to physical systems. The ultimate goal is to provide an efficient solution that compresses the space in which the operator operates, analogous to PCA for matrices.

Data

1. MacCallum S, Merchant C (2011) Arc-lake v1.1-per-lake, 1995–2009 [1](https://doi.org/10.7488/ds/159)
2. Reiss, A. (2012). PAMAP2 Physical Activity Monitoring [Dataset]. UCI Machine Learning Repository. [2](https://doi.org/10.24432/C5NW2H)
3. Bousseljot R, Kreiseler D, Schnabel, A. Nutzung der EKG-Signaldatenbank CARDIODAT der PTB über das Internet. Biomedizinische Technik, Band 40, Ergänzungsband 1 (1995) S 317[3](https://doi.org/10.13026/C28C71)

Code

1. https://github.com/ilkhem/icebeem
2. https://github.com/ilkhem/ivae
3. https://github.com/kondratevakate/fmri-component-analysis/blob/master/VAE_for_fMRI/dataset/train/Bystrova0_y-axis.png

References

1. Kovachki, Nikola B., Samuel Lanthaler, and Andrew M. Stuart. "Operator learning: Algorithms and analysis." arXiv preprint arXiv:2402.15715 (2024). [4](https://doi.org/10.48550/arXiv.2402.15715)
2. Palummo, A., Arnone, E., Formaggia, L. et al. Functional principal component analysis for incomplete space–time data. Environ Ecol Stat 31, 555–582 (2024). [5](https://doi.org/10.1007/s10651-024-00598-7)
3. Gruber, Anthony, and Irina Tezaur. "Canonical and noncanonical Hamiltonian operator inference." Computer Methods in Applied Mechanics and Engineering 416 (2023): 116334. [6](https://doi.org/10.1016/j.cma.2023.116334)
4. Olivieri, M., Karakonstantis, X., Pezzoli, M. et al. Physics-informed neural network for volumetric sound field reconstruction of speech signals. J AUDIO SPEECH MUSIC PROC. 2024, 42 (2024). [7](https://doi.org/10.1186/s13636-024-00366-2)

Baseline

Apply The Fourier operator or other similar operator and use this transformation to predict the time series.

Proposed solution

Using an operator learning framework, propose a solution for different types of operators used in physics, image analysis, etc. Propose an idea to implement ICA to reduce the dimensionality of an operator to compute it in an efficient way.

Novelity

There are no solutions to learning the operators used in physics.

Authors

• Alexander Terentyev