Samuel Lanthaler

I am a postdoctoral researcher and visiting scholar at Caltech, where I am hosted by Andrew Stuart.

My current research focuses on

numerical methods for incompressible fluid flows
novel neural network-based methods for operator approximation
and applications of such methods to Bayesian inversion

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.

News

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)!
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).

Curriculum vitae

since 2022	Postdoc at Caltech
2021 – 2022	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)

My detailed CV; mathematics thesis; physics thesis.

Teaching

Spring 2022: Lecturer for course on Numerical methods for hyperbolic PDEs at ETH Zurich

Publications & Preprints | Google Scholar | arXiv |

Samuel Lanthaler and Andrew M. Stuart,
The curse of dimensionality in operator learning.
Preprint, arXiv:2306.15924 (2023)
[ preprint | bib ]

@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.}
}

Samuel Lanthaler and Nicholas H. Nelsen,
Error Bounds for Learning with Vector-Valued Random Features.
Preprint, arXiv:2305.17170 (2023)
[ preprint | bib ]

@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.}
}

Samuel Lanthaler, T. Konstantin Rusch and Siddhartha Mishra,
Neural Oscillators are Universal.
Preprint, arXiv:2305.08753 (2023)
[ preprint | bib ]

@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 }
}

Samuel Lanthaler, Zongyi Li and Andrew M. Stuart,
The Nonlocal Neural Operator: Universal Approximation.
Preprint, arXiv:2304.13221 (2023)
[ preprint | bib ]

@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.}
}

Samuel Lanthaler,
Operator learning with PCA-Net: upper and lower complexity bounds.
Preprint, arXiv:2303.16317 (2023)
[ preprint | bib ]

@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}
}

Samuel Lanthaler,
On concentration in vortex sheets.
Partial Differ. Equ. Appl., 4(13) (2023)
[ url | preprint | bib ]

@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}
}

Samuel Lanthaler, Roberto Molinaro, Patrik Hadorn and Siddhartha Mishra,
Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities.
In 11th International Conference on Learning Representations (ICLR) (2023)
[ pdf | preprint | bib ]

@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 }
}

Samuel Lanthaler, Siddhartha Mishra and Franziska Weber,
On Bayesian data assimilation for PDEs with ill-posed forward problems.
Inverse Problems, 38(8):085012 (2022)
[ url | preprint | bib ]

@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 }
}

Samuel Lanthaler, Siddhartha Mishra and George E Karniadakis,
Error estimates for DeepONets: A deep learning framework in infinite dimensions.
Transactions of Mathematics and Its Applications, 6(1):tnac001 (2022)
[ url | preprint | bib ]

@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}
}

Nikola Kovachki, Samuel Lanthaler and Siddhartha Mishra,
On Universal Approximation and Error Bounds for Fourier Neural Operators.
Journal of Machine Learning Research, 22(290):1-76 (2021)
[ url | pdf | preprint | bib ]

@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 }
}

Tim De Ryck, Samuel Lanthaler and Siddhartha Mishra,
On the approximation of functions by tanh neural networks.
Neural Networks, 143:732-750 (2021)
[ url | preprint | bib ]

@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 }
}

Samuel Lanthaler, Siddhartha Mishra and Carlos Parés-Pulido,
Statistical solutions of the incompressible Euler equations.
Mathematical Models and Methods in Applied Sciences, 31(2):223-292 (2021)
[ url | preprint | bib ]

@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 }
}

Samuel Lanthaler, Siddhartha Mishra and Carlos Parés-Pulido,
On the conservation of energy in two-dimensional incompressible flows.
Nonlinearity, 34(2):1084 (2021)
[ url | preprint | bib ]

@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 }
}

Samuel Lanthaler and Siddhartha Mishra,
On the convergence of the spectral viscosity method for the two-dimensional incompressible Euler equations with rough initial data.
Foundations of Computational Mathematics, 20(5):1309-1362 (2020)
[ url | preprint | bib ]

@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 }
}

Samuel Lanthaler, Jonathan P Graves, David Pfefferlé and Wilfred Anthony Cooper,
Guiding-centre theory for kinetic-magnetohydrodynamic modes in strongly flowing plasmas.
Plasma Physics and Controlled Fusion, 61(7):074006 (2019)
[ url | bib ]

@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}
}

S Lanthaler, D Pfefferlé, JP Graves and WA Cooper,
Higher order Larmor radius corrections to guiding-centre equations and application to fast ion equilibrium distributions.
Plasma Physics and Controlled Fusion, 59(4):044014 (2017)
[ url | bib ]

@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 }
}

Ulrik Skre Fjordholm, Samuel Lanthaler and Siddhartha Mishra,
Statistical solutions of hyperbolic conservation laws: foundations.
Archive for Rational Mechanics and Analysis, 226(2):809-849 (2017)
[ url | preprint | bib ]

@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 }
}

Samuel Lanthaler and Siddhartha Mishra,
Computation of measure-valued solutions for the incompressible Euler equations.
Mathematical Models and Methods in Applied Sciences, 25(11):2043-2088 (2015)
[ url | preprint | bib ]

@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 }
}

My complete publication list, including publications as second(++) author.

Office
315 Annenberg IST Center

Address
California Institute of Technology
1200 E. California Blvd., MC 305-16
Pasadena, CA 91125

Contact

Samuel Lanthaler

Computing + Mathematical Sciences

California Institute of Technology

News

Curriculum vitae

Teaching

Publications & Preprints | Google Scholar | arXiv |