I am a postdoctoral researcher and visiting scholar at Caltech, where I am hosted by Andrew Stuart.
My current research focuses on
For the duration of my stay at Caltech, I have been awarded a Postdoc.Mobility grant by the Swiss National Science Foundation (2022-2024).
I have previously been a postdoc and lecturer at the Seminar for Applied Mathematics, ETH Zurich, where I also completed my PhD in mathematics under the supervision of Siddhartha Mishra.
Oct 2023 | Our two submissions to NeurIPS 2023 got accepted, Error Bounds for Learning with Vector-Valued Random Features (spotlight) and Neural Oscillators are Universal (poster)! |
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June 2023 | New work addresses the The curse of dimensionality in operator learning (joint with Andrew Stuart). We show rigorously that operator learning on general classes of (e.g. Lipschitz continuous) operators, suffers from a curse of dimensionality: to achieve a desired accuracy epsilon, operator learning models generally need to be exponentially large in the inverse of the accuracy. Our result covers DeepONets, Fourier neural operators and PCA-Net, among others. |
May 2023 | Our new preprint on Error Bounds for Learning with Vector-Valued Random Features (joint with Nick Nelsen) develops theory for operator learning with random features, including quantitative convergence rates and general asymptotic consistency results. |
May 2023 | Oscillatory systems are fundamental in physics and biology. They have also found their way into ML via oscillator-based neural ODEs; We prove that Neural Oscillators are Universal (joint with T. Konstantin Rusch and Sid Mishra). |
Postdoc at Caltech | |
Postdoc/Lecturer at ETH Zurich | |
2018 – 2021 | PhD in Mathematics at ETH Zurich (under the supervision of Prof. S. Mishra) |
2015 – 2020 | PhD in Physics at EPF Lausanne (under the supervision of Prof. J.P. Graves) |
2013 – 2015 | MSc in Mathematics (ETH Zurich) |
2010 – 2013 | BSc in Mathematics (ETH Zurich) |
@misc{lanthaler2023curse, title = {The curse of dimensionality in operator learning}, year = {2023}, eprint = {2306.15924}, archivePrefix = {arXiv}, primaryClass = {cs.LG}, author = {Lanthaler, Samuel and Stuart, Andrew M.} }
@misc{lanthaler2023error, title = {Error Bounds for Learning with Vector-Valued Random Features}, year = {2023}, eprint = {2305.17170}, archivePrefix = {arXiv}, primaryClass = {stat.ML}, author = {Lanthaler, Samuel and Nelsen, Nicholas H.} }
@misc{lanthaler2023neural, title = {Neural Oscillators are Universal}, year = {2023}, eprint = {2305.08753}, archivePrefix = {arXiv}, primaryClass = {cs.NE}, author = {Lanthaler, Samuel and Rusch, T. Konstantin and Mishra, Siddhartha } }
@misc{LLS2023, title = {The Nonlocal Neural Operator: Universal Approximation}, year = {2023}, eprint = {2304.13221}, archivePrefix = {arXiv}, primaryClass = {math.NA}, author = {Lanthaler, Samuel and Li, Zongyi and Stuart, Andrew M.} }
@misc{lanthaler2023operator, title = {Operator learning with PCA-Net: upper and lower complexity bounds}, year = {2023}, eprint = {2303.16317}, archivePrefix = {arXiv}, primaryClass = {cs.LG}, author = {Samuel Lanthaler} }
@article{lanthaler2023concentration, title = {On concentration in vortex sheets}, journal = {Partial Differ. Equ. Appl.}, volume = {4}, number = {13}, url = {https://link.springer.com/article/10.1007/s42985-023-00230-6}, year = {2023}, publisher = {Springer}, eprint = {2004.01537}, author = {Samuel Lanthaler} }
@inproceedings{LMHM2022, abbr = {ICLR}, title = {Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities}, booktitle = {11th International Conference on Learning Representations (ICLR)}, publisher = {OpenReview.net}, year = {2023}, pdf = {https://openreview.net/forum?id=CrfhZAsJDsZ}, eprint = {2210.01074}, author = {Lanthaler, Samuel and Molinaro, Roberto and Hadorn, Patrik and Mishra, Siddhartha } }
@article{LMW2022, doi = {10.1088/1361-6420/ac7acd}, url = {https://doi.org/10.1088/1361-6420/ac7acd}, year = {2022}, month = {7}, publisher = {{IOP} Publishing}, volume = {38}, number = {8}, pages = {085012}, eprint = {2107.07593}, title = {On {B}ayesian data assimilation for {PDEs} with ill-posed forward problems}, journal = {Inverse Problems}, abstract = {We study Bayesian data assimilation (filtering) for time-evolution Partial differential equations (PDEs), for which the underlying forward problem may be very unstable or ill-posed. Such PDEs, which include the Navier–Stokes equations of fluid dynamics, are characterized by a high sensitivity of solutions to perturbations of the initial data, a lack of rigorous global well-posedness results as well as possible non-convergence of numerical approximations. Under very mild and readily verifiable general hypotheses on the forward solution operator of such PDEs, we prove that the posterior measure expressing the solution of the Bayesian filtering problem is stable with respect to perturbations of the noisy measurements, and we provide quantitative estimates on the convergence of approximate Bayesian filtering distributions computed from numerical approximations. For the Navier–Stokes equations, our results imply uniform stability of the filtering problem even at arbitrarily small viscosity, when the underlying forward problem may become ill-posed, as well as the compactness of numerical approximants in a suitable metric on time-parametrized probability measures.}, author = {Lanthaler, Samuel and Mishra, Siddhartha and Weber, Franziska } }
@article{lanthaler2022error, title = {Error estimates for {D}eep{ON}ets: {A} deep learning framework in infinite dimensions}, journal = {Transactions of Mathematics and Its Applications}, volume = {6}, number = {1}, pages = {tnac001}, url = {https://academic.oup.com/imatrm/article-pdf/6/1/tnac001/42785544/tnac001.pdf}, year = {2022}, publisher = {Oxford University Press}, eprint = {2102.09618}, author = {Lanthaler, Samuel and Mishra, Siddhartha and Karniadakis, George E} }
@article{KLM_JMLR2020, title = {On Universal Approximation and Error Bounds for Fourier Neural Operators}, journal = {Journal of Machine Learning Research}, year = {2021}, volume = {22}, number = {290}, pages = {1-76}, url = {http://jmlr.org/papers/v22/21-0806.html}, pdf = {https://jmlr.org/papers/volume22/21-0806/21-0806.pdf}, eprint = {2107.07562}, author = {Kovachki, Nikola and Lanthaler, Samuel and Mishra, Siddhartha } }
@article{DERYCK2021732, title = {On the approximation of functions by tanh neural networks}, journal = {Neural Networks}, volume = {143}, pages = {732-750}, year = {2021}, issn = {0893-6080}, doi = {https://doi.org/10.1016/j.neunet.2021.08.015}, url = {https://www.sciencedirect.com/science/article/pii/S0893608021003208}, eprint = {2104.08938}, author = {Ryck, Tim De and Lanthaler, Samuel and Mishra, Siddhartha } }
@article{LMPP2021, title = {Statistical solutions of the incompressible Euler equations}, journal = {Mathematical Models and Methods in Applied Sciences}, volume = {31}, number = {02}, pages = {223-292}, year = {2021}, doi = {10.1142/S0218202521500068}, eprint = {1909.06615}, author = {Lanthaler, Samuel and Mishra, Siddhartha and Parés-Pulido, Carlos } }
@article{LMPP2021a, title = {On the conservation of energy in two-dimensional incompressible flows}, journal = {Nonlinearity}, volume = {34}, number = {2}, pages = {1084}, year = {2021}, doi = {https://dx.doi.org/10.1088/1361-6544/abb452}, publisher = {IOP Publishing}, eprint = {2001.06195}, author = {Lanthaler, Samuel and Mishra, Siddhartha and Parés-Pulido, Carlos } }
@article{LM2020, title = {On the convergence of the spectral viscosity method for the two-dimensional incompressible Euler equations with rough initial data}, journal = {Foundations of Computational Mathematics}, volume = {20}, number = {5}, pages = {1309--1362}, year = {2020}, publisher = {Springer}, doi = {https://doi.org/10.1007/s10208-019-09440-0}, eprint = {1903.12361}, author = {Lanthaler, Samuel and Mishra, Siddhartha } }
@article{lanthaler2019guiding, title = {Guiding-centre theory for kinetic-magnetohydrodynamic modes in strongly flowing plasmas}, journal = {Plasma Physics and Controlled Fusion}, volume = {61}, number = {7}, pages = {074006}, year = {2019}, publisher = {IOP Publishing}, doi = {https://doi.org/10.1088/1361-6587/aa5e70}, author = {Lanthaler, Samuel and Graves, Jonathan P and Pfefferlé, David and Cooper, Wilfred Anthony} }
@article{lanthaler2017higher, title = {Higher order Larmor radius corrections to guiding-centre equations and application to fast ion equilibrium distributions}, journal = {Plasma Physics and Controlled Fusion}, volume = {59}, number = {4}, pages = {044014}, year = {2017}, publisher = {IOP Publishing}, doi = {https://doi.org/10.1088/1361-6587/ab1d21}, author = {Lanthaler, S and Pfefferlé, D and Graves, JP and Cooper, WA } }
@article{fjordholm2017statistical, title = {Statistical solutions of hyperbolic conservation laws: foundations}, journal = {Archive for Rational Mechanics and Analysis}, volume = {226}, number = {2}, pages = {809--849}, year = {2017}, publisher = {Springer}, doi = {https://doi.org/10.1007/s00205-017-1145-9}, eprint = {1605.05960}, author = {Fjordholm, Ulrik Skre and Lanthaler, Samuel and Mishra, Siddhartha } }
@article{LM2015, title = {Computation of measure-valued solutions for the incompressible Euler equations}, journal = {Mathematical Models and Methods in Applied Sciences}, volume = {25}, number = {11}, pages = {2043--2088}, year = {2015}, publisher = {World Scientific}, doi = {https://doi.org/10.1142/S0218202515500529}, eprint = {1411.5064}, author = {Lanthaler, Samuel and Mishra, Siddhartha } }