Samuel Lanthaler

Computing + Mathematical Sciences

California Institute of Technology


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

April 2024 Our new preprint gives an overview of theoretical aspects of operator learning. This is joint with Nik Kovachki and Andrew Stuart, Operator Learning: Algorithms and Analysis.
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)!

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

Publications & Preprints   | Google Scholar | arXiv |

  1. Nikola Kovachki, Samuel Lanthaler and Andrew M. Stuart,
    Operator Learning: Algorithms and Analysis.
    Preprint, arXiv:2402.15715 (2024)
    [ preprint | ] 
    @article{kovachki2024operator,
    title = {Operator {Learning}: Algorithms and {Analysis}},
    year = {2024},
    eprint = {2402.15715},
    archivePrefix = {arXiv},
    author = {Kovachki, Nikola  and Lanthaler, Samuel  and Stuart, Andrew M.}
}
  2. Samuel Lanthaler and Andrew M. Stuart,
    The Parametric Complexity of Operator Learning.
    Preprint, arXiv:2306.15924 (2024)
    [ preprint | ] 
    @misc{lanthaler2024parametric,
    title = {The Parametric Complexity of Operator Learning},
    year = {2024},
    eprint = {2306.15924},
    archivePrefix = {arXiv},
    primaryClass = {cs.LG},
    author = {Lanthaler, Samuel  and Stuart, Andrew M.}
}
  3. Samuel Lanthaler and Nicholas H. Nelsen,
    Error Bounds for Learning with Vector-Valued Random Features.
    In Advances in Neural Information Processing Systems (2023)
    [ url | ] 
    @inproceedings{LN2023randomfeatures,
    booktitle = {Advances in Neural Information Processing Systems},
    pages = {71834--71861},
    publisher = {Curran Associates, Inc.},
    title = {Error Bounds for Learning with Vector-Valued Random Features},
    url = {https://proceedings.neurips.cc/paper\_files/paper/2023/file/e34d908241aef40440e61d2a27715424-Paper-Conference.pdf},
    volume = {36},
    year = {2023},
    author = {Lanthaler, Samuel  and Nelsen, Nicholas H.}
}
  4. Samuel Lanthaler, T. Konstantin Rusch and Siddhartha Mishra,
    Neural Oscillators are Universal.
    In Advances in Neural Information Processing Systems (2023)
    [ url | ] 
    @inproceedings{lanthaler2023neuraloscillators,
    booktitle = {Advances in Neural Information Processing Systems},
    pages = {46786--46806},
    publisher = {Curran Associates, Inc.},
    title = {Neural Oscillators are Universal},
    url = {https://proceedings.neurips.cc/paper\_files/paper/2023/file/923285deb805c3e14e1aeebc9854d644-Paper-Conference.pdf},
    volume = {36},
    year = {2023},
    author = {Lanthaler, Samuel  and Rusch, T. Konstantin and Mishra, Siddhartha }
}
  5. Samuel Lanthaler, Zongyi Li and Andrew M. Stuart,
    The Nonlocal Neural Operator: Universal Approximation.
    Preprint, arXiv:2304.13221 (2023)
    [ preprint | ] 
    @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.}
}
  6. Samuel Lanthaler,
    Operator learning with PCA-Net: upper and lower complexity bounds.
    Journal of Machine Learning Research, 24(318):1-67 (2023)
    [ url | ] 
    @article{lanthaler2023pcanet,
    title = {Operator learning with PCA-Net: upper and lower complexity bounds},
    journal = {Journal of Machine Learning Research},
    year = {2023},
    volume = {24},
    number = {318},
    pages = {1--67},
    url = {http://jmlr.org/papers/v24/23-0478.html},
    author = {Samuel  Lanthaler}
}
  7. Samuel Lanthaler,
    On concentration in vortex sheets.
    Partial Differ. Equ. Appl., 4(13) (2023)
    [ url | preprint | ] 
    @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}
}
  8. 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 | ] 
    @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 }
}
  9. 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 | ] 
    @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 }
}
  10. 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 | ] 
    @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}
}
  11. 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 | ] 
    @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 }
}
  12. 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 | ] 
    @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 }
}
  13. 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 | ] 
    @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 }
}
  14. 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 | ] 
    @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 }
}
  15. 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 | ] 
    @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 }
}
  16. 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 | ] 
    @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}
}
  17. 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 | ] 
    @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 }
}
  18. 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 | ] 
    @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 }
}
  19. 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 | ] 
    @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.
personal pic

Office
315 Annenberg IST Center
Address
California Institute of Technology
1200 E. California Blvd., MC 305-16
Pasadena, CA 91125
Contact