Research

My research path was initially somewhat eclectic. During my undergraduate studies, I took a mix of pure and applied math electives in my fourth year, and I wrote my final degree project on the Riemann zeta function and its relationship to the distribution of prime numbers. Later, I decided to transition away from pure mathematics, and my master's thesis served as my first introduction to the finite element method—a topic I had not studied at all during my undergraduate degree. Since then, I have fully immersed myself in numerical analysis to solve partial differential equations, with a special focus on neural networks.

Below is my research summary and a record detailing my publications, delivered talks, participation in projects, and academic service.

Summary

One of the major concerns in the numerical solution of partial differential equations is achieving an optimal transition from the continuum framework—where the PDE is posed, typically involving infinite-dimensional Banach or Hilbert spaces—to the discrete setup—where the solution approximation is found, traditionally in the form of finite-dimensional subspaces of those at the continuum level.

The finite element method is one of the most established approaches in this regard. The discrete basis functions are constructed over a mesh imposed on the computational domain, with local support and piecewise-polynomial shapes to exploit matrix sparsity for computational efficiency and control numerical integration. This method has been a long-standing cornerstone of science and engineering and remains one of the most robust tools for simulating complex physical systems. However, its most notorious limitation is the curse of dimensionality—the exponential explosion in computational cost and memory requirements as the number of spatial or parametric dimensions increases.

This was the point at which my PhD research began. I became interested in whether neural networks and deep learning could offer a different way to approximate PDE solutions without traditional mesh-based machinery. As I went deeper into these methods, however, I became increasingly aware of the gap between their promise in scientific computing and what could actually be justified mathematically. Rather than viewing deep learning as a universal black-box solution, I became more interested in understanding where these methods work, where they fail, and what mathematical structure is needed to make them reliable. This shift has shaped my research ever since, moving me from simply applying new tools to established problems towards identifying the underlying mathematical questions and developing methods around them. My current work lies at the intersection of numerical analysis and machine learning, with a focus on neural-network-based methods for PDEs, residual minimization, variational formulations, and adaptivity.

PhD Dissertation

"Solving partial differential equations using artificial neural networks / Deribatu partzialetako ekuazioen ebazpena neurona-sare artifizialen bidez"

• For a revised English-only version, find it here.

• For the original, bilingual version—containing some minor errata identified after the defense, find it here.

In 2025, this thesis was recognized as the best PhD thesis defended in 2024 by the Spanish Society for Applied Mathematics (SEMA), the Spanish Society of Numerical Methods in Engineering (SEMNI), and the European Community in Computational Methods in Applied Sciences (ECCOMAS).

Publications

[11] Alejandro Duque, Paulina Sepúlveda, Carlos Uriarte, Jamie M. Taylor, and David Pardo. “Robust Deep FOSLS for Transmission Problems”. In: Computer Methods in Applied Mechanics and Engineering 461 (2026), p. 119236. issn: 0045-7825. doi: https://doi.org/10.1016/j.cma.2026.119236.

[10] Tomasz Slużalec, Daria Wójcik, Carlos Uriarte, Marcin Loś, Anna Paszyńska, and Maciej Paszyński. “Reliable physics-informed neural networks for Navier–Stokes simulations. Can we trust AI-generated numerical simulations?”. In: Journal of Computational Science 95 (2026), p. 102817. issn: 1877-7503. doi: https://doi.org/10.1016/j.jocs.2026.102817.

[9] Danilo Aballay, Federico Fuentes, Vicente Iligaray, Angel J. Omella, David Pardo, Manuel A. Sánchez, Ignacio Tapia, and Carlos Uriarte. “An r-adaptive finite element method using neural networks for parametric self-adjoint elliptic problems”. In: Journal of Computational Physics 545 (2026), p. 114447. issn: 0021-9991. doi: https://doi.org/10.1016/j.jcp.2025.114447.

[8] Askold Vilkha, Carlos Uriarte, Pawel Maczuga, Tomasz Slużalec, and Maciej Paszyński. “Tensorial Implementation for Robust Variational Physics-Informed Neural Networks”. In: Computational Science – ICCS 2025. Ed. by Michael H. Lees, Wentong Cai, Siew Ann Cheong, Yi Su, David Abramson, Jack J. Dongarra, and Peter M. A. Sloot. Cham: Springer Nature Switzerland, 2025, pp. 61–75. isbn: 978-3-031-97626-1. doi: https://doi.org/10.1007/978-3-031-97626-1_5.

[7] Shima Baharlouei, Jamie M. Taylor, Carlos Uriarte, and David Pardo. “A Least-Squares-Based Neural Network (LS-Net) for Solving Linear Parametric PDEs”. In: Computer Methods in Applied Mechanics and Engineering 437 (2025), p. 117757. issn: 0045-7825. doi: https://doi.org/10.1016/j.cma.2025.117757.

[6] Marcin Loś, Tomasz Slużalec, Pawel Maczuga, Askold Vilkha, Carlos Uriarte, and Maciej Paszyński. “Collocation-Based Robust Variational Physics-Informed Neural Networks (CRVPINN)”. In: Computers & Structures 316 (2025), p. 107839. issn: 0045-7949. doi: https://doi.org/10.1016/j.compstruc.2025.107839.

[5] Carlos Uriarte, Manuela Bastidas, David Pardo, Jamie M. Taylor, and Sergio Rojas. “Optimizing Variational Physics-Informed Neural Networks Using Least Squares”. In: Computers & Mathematics with Applications 185 (2025), pp. 76–93. issn: 0898-1221. doi: https://doi.org/10.1016/j.camwa.2025.02.022.

[4] Mateusz Dobija, Anna Paszyńska, Carlos Uriarte, and Maciej Paszyński. “Accelerating Training of Physics-Informed Neural Network with Hierarchical Matrices”. In: Computational Science – ICCS 2024. Ed. by Leonardo Franco, Clélia de Mulatier, Maciej Paszynski, Valeria V. Krzhizhanovskaya, Jack J. Dongarra, and Peter M. A. Sloot. Cham: Springer Nature Switzerland, 2024, pp. 352–362. isbn: 978-3-031-63759-9. doi: https://doi.org/10.1007/978-3-031-63759-9_38.

[3] Carlos Uriarte, Jamie M. Taylor, David Pardo, Oscar A. Rodríguez, and Patrick Vega. “Memory-Based Monte Carlo Integration for Solving Partial Differential Equations Using Neural Networks”. In: Computational Science – ICCS 2023. Ed. by Jiří Mikyška, Clélia de Mulatier, Maciej Paszynski, Valeria V. Krzhizhanovskaya, Jack J. Dongarra, and Peter M.A. Sloot. Cham: Springer Nature Switzerland, 2023, pp. 509–516. isbn: 978-3-031-36021-3. doi: https://doi.org/10.1007/978-3-031-36021-3_51.

[2] Carlos Uriarte, David Pardo, Ignacio Muga, and Judit Muñoz-Matute. “A Deep Double Ritz Method (D2RM) for Solving Partial Differential Equations Using Neural Networks”. In: Computer Methods in Applied Mechanics and Engineering 405 (2023), p. 115892. issn: 0045-7825. doi: https://doi.org/10.1016/j.cma.2023.115892.

[1] Carlos Uriarte, David Pardo, and Angel J. Omella. “A Finite Element Based Deep Learning Solver for Parametric PDEs”. In: Computer Methods in Applied Mechanics and Engineering 391 (2022), p. 114562. issn: 0045-7825. doi: https://doi.org/10.1016/j.cma.2021.114562.

Under review / in progress

• Carlos Uriarte and Víctor M. Calo. "A minimum-residual multiscale Galerkin realization toward stabilized adaptivity".

• Pablo Herrera, Jamie M. Taylor, Carlos Uriarte, Ignacio Μuga, David Pardo, Kristoffer G. Van Der Zee. "RUNNs: Ritz-Uzawa Neural Networks for Solving Variational Problems".

• Víctor M. Calo and Carlos Uriarte. "A minimum-residual multiscale Galerkin framework for a-posteriori eigenpair reliability".

Talks

[27] Oral presentation: “A minimum-residual multiscale Galerkin framework with applications on h- and r-adaptivity using neural networks”. In: 13th Edition of the Biennial Congress on Numerical Methods (CMN), Gijón, Spain, July 2026. 

[26] Plenary talk: “Solving Partial Differential Equations Using Artificial Neural Networks”. In: XXI Jacques-Louis Lions Hispano-French School on Numerical Simulation in Physics and Engineering (EHF 2025), Ciudad Real, Spain, July 2025. 

[25] Seminar: “Solving Partial Differential Equations Using Artificial Neural Networks”. In: Institute of Mathematics of the Pontificia Universidad Católica de Valparaíso, Chile, July 2025. 

[24] Oral presentation: “Solving Parametric Partial Differential Equations Using Least-Squares-Based Neural Networks”. In: 3rd IACM Digital Twins in Engineering Conference (DTE 2025) & 1st ECCOMAS Artificial Intelligence and Computational Methods in Applied Sciences (AICOMAS 2025), Paris, France, Feb. 2025. 

[23] Seminar: “Solving Parametric Partial Differential Equations Combining Artificial Neural Networks and Least Squares”. In: Department of Mathematics of the Faculty of Sciences of the National University of Colombia (UNAL), Medellín Campus, Colombia, Nov. 2024. 

[22] Seminar: “Solving Parametric Partial Differential Equations Combining Artificial Neural Networks and Least Squares”. In: Department of Mathematics of the Faculty of Sciences of the National University of Colombia (UNAL), Bogotá Campus, Colombia, Nov. 2024. 

[21] Oral presentation: “Maximizing Opportunities in Congresses, Seminars, and Workshops”. In: Transversal Skills Program of the 1st In-Deep Week, Bilbao, Spain, Oct. 2024. 

[20] Oral presentation: “Goal-Oriented Adaptivity for Solving Partial Differential Equations Using Artificial Neural Networks”. In: 16th World Congress on Computational Mechanics and 4th Pan American Congress on Computational Mechanics (WCCM/PANACM 2024), Vancouver, British Columbia, Canada, July 2024. 

[19] Oral presentation: “Goal-Oriented Adaptivity Using Artificial Neural Networks”. In: 24th International Conference on Computational Science (ICCS 2024), Malaga, Spain, July 2024. 

[18] Oral presentation: “Optimizing Variational Physics-Informed Neural Networks Using Least Squares”. In: 6th Edition of the Minimum Residual & Least-Squares Finite Element Methods (MINRES) Workshop, Bilbao, Spain, June 2024. 

[17] Oral presentation: “Optimizing Variational Physics-Informed Neural Networks Using Least Squares”. In: XXVIII Congress of Differential Equations and Applications / XVIII Congress of Applied Mathematics (CEDYA/CMA 2024), Bilbao, Spain, June 2024. 

[16] Oral presentation: “Goal-Oriented Adaptivity for Solving Partial Differential Equations Using Neural Networks”. In: 9th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2024), Lisbon, Portugal, June 2024. 

[15] Oral presentation: “The Deep Double Ritz Method: A Deep Learning Residual Method for Solving Partial Differential Equations”. In: 2nd IACM Mechanistic Machine Learning and Digital Engineering for Computational Science and Technology Conference, University of Texas at El Paso, USA, Sept. 2023. 

[14] Oral presentation: “Goal-oriented Deep Ritz and Least-Squares methods”. In: 23rd International Conference on Computational Science (ICCS 2023), Prague, Czech Republic, July 2023. 

[13] Oral presentation: “Memory-based Monte Carlo integration for solving Partial Differential Equations using Neural Networks”. In: 23rd International Conference on Computational Science (ICCS 2023), Prague, Czech Republic, July 2023. 

[12] Seminar: “(Goal-Oriented) Deep Residual Minimization Methods”. In: Faculty of Computer Science, Electronics, and Telecommunications of the AGH University of Science and Technology in Krakow, Poland, July 2023. 

[11] Oral presentation: “A Deep Double Ritz Method for solving Partial Differential Equations using Neural Networks”. In: Workshop Numerical Methods in Geophysics: Present, Future, and Applications, Valparaíso, Chile, Jan. 2023. 

[10] Oral presentation: “A Deep Double Ritz Method for solving Partial Differential Equations”. In: XC Encuentro Anual de la Sociedad Matemática de Chile, Punta de Tralca, Chile, Dec. 2022. 

[9] Seminar: “A Deep Double Ritz Method for solving Partial Differential Equations”. In: Institute of Mathematics of the Pontificia Universidad Católica de Valparaíso, Chile, Nov. 2022. 

[8] Oral presentation: “Solving Partial Differential Equations using Adversarial Neural Networks”. In: Congress on Numerical Methods in Engineering (CMN 2022), Las Palmas de Gran Canaria, Spain, Sept. 2022. 

[7] Oral presentation: “A Finite Element based Deep Learning solver for parametric PDEs”. In: SIAM Conference on Mathematics of Data Science (MDS 22), San Diego, California, USA, Sept. 2022. 

[6] Oral presentation: “Adversarial Neural Networks for solving variationally formulated Partial Differential Equations”. In: 15th World Congress on Computational Mechanics and 8th Asian Pacific Congress on Computational Mechanics (WCCM/APCOM 2022), Yokohama, Japan, July 2022. 

[5] Oral presentation: “A Generative Adversarial Networks approach for solving Partial Differential Equations”. In: 8th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2022), Oslo, Norway, June 2022. 

[4] Oral presentation: “An Adversarial Networks approach for solving Partial Differential Equations”. In: 22nd International Conference on Computational Science (ICCS 2022), London, UK, June 2022. 

[3] Oral presentation: “A Finite Element based Deep Learning solver for parametric PDEs”. In: 21st International Conference on Computational Science (ICCS 2021), Krakow, Poland, June 2021. 

[2] Seminar: “A Finite Element based Deep Learning solver for parametric PDEs”. In: Institute of Mathematics of the Pontificia Universidad Católica de Valparaíso, Chile, Mar. 2021. 

[1] Seminar: “A Finite Element based Deep Learning solver for parametric PDEs”. In: Faculty of Computer Science, Electronics, and Telecommunications of the AGH University of Science and Technology in Krakow, Poland, Jan. 2021.

Projects

• 2026–2030: “MSCA Staff Exchange project (MOST-PINN)” (Ref. No.: 101342459). European Commission (“Horizon Europe Marie Skłodowska-Curie Actions Staff Exchanges, HORIZON-MSCA-SE”). Principal Investigator (EHU): David Pardo. Amount awarded: 868,560€.

• 2025–2027: “Postdoctoral scholarship” (Ref. No.: POS-2024-1-0004). Basque Government (“Ayudas para el Programa Posdoctoral, de Perfeccionamiento de Personal Investigador Doctor, correspondiente al periodo 2024-2027”). Principal Investigators: Carlos Uriarte,  David Pardo. Amount awarded: 136,123€. Note: Received for 19 months. Waived in favor of a professorship, which could not be held concurrently.

• 2024–2028: “UltraPINNs to improve imaging in diagnostic (ULTRAPINNS)” (Ref. No.: PID2023-146678OB-I00). Spanish Ministry of Science, Innovation and Universities. State Research Agency. Principal Investigators: David Pardo, Vincenzo Nava. Amount awarded: 250,000€.

• 2024–2028: “Deep learning techniques for quantification of uncertainty in infinite-dimensional inversion and patient characterization of brain fluid motion (BRAINVERSE)” (Ref. No.: PID2023-146668OA-I00). Spanish Ministry of Science, Innovation and Universities. State Research Agency. Principal Investigators: Matteo Croci, Judit Muñoz-Matute. Amount awarded: 60,000€.

• 2024–2026: “Remaining useful life of mooring lines and umbilicals for offshore renewables (RUL-ET)” (Ref. No.: KK-2024/00086). Basque Government (“ELKARTEK”). Amount awarded: 552,000€ (Tecnalia) and 104,000€ (BCAM).

• 2022–2025: “Recognition of MATHMODE as a Research Group of Excellence (A+)” (Ref. No.: IT1456-22). Basque Government (“Ayudas para apoyar las actividades de grupos de investigación del sistema universitario vasco para los años 2022-2025”). Principal Investigators: David Pardo, Inmaculada Arostegui. Amount awarded: 486,150€.

• 2022–2024: “Advanced Numerical Methods and Neural Networks for the Structural Health of Offshore Wind Platforms (MATHEOLO)” (Ref. No.: TED2021-132783B-I00). Spanish Ministry of Science, Innovation and Universities. State Research Agency. Principal Investigators: Vincenzo Nava, David Pardo. Amount awarded: 147,200€.

• 2021–2024: “New proposals for estimation, prediction, and validation of semi-parametric models for the analysis of complex data with applications in Health and Climate Change” (Ref. No.: PID2020-115882RB-I00). Spanish Ministry of Science, Innovation and Universities. State Research Agency. Principal Investigator: Dae-Jin Lee. Amount awarded: 38,115€.

• 2020–2024: “Predoctoral Scholarship” (Ref. No.: SEV-2017-0718-19-2/PRE2019-087878). Spanish Ministry of Science and Innovation. State Research Agency (“Ayudas para contratos predoctorales para la formación de doctores/as 2019 del Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020”). Principal Investigators: Luis Vega, David Pardo. Amount awarded: 98,200€.

• 2020–2024: “Predoctoral Scholarship” (Ref. No.: PIF19/101). University of the Basque Country (“Convocatoria de contratación para la formación de personal investigador en la UPV/EHU 2019”). Principal Investigators: David Pardo, Elisabete Alberdi. Amount awarded: 85,714€. Note: Received for 7 months. Waived in favor of the above, which could not be held concurrently.

• 2020–2023: “Real-time Inversion using Deep Learning Methods (DEEPINVERSE)” (Ref. No.: PID2019-108111RB-I00). Spanish Ministry of Science, Innovation and Universities. State Research Agency. Principal Investigator: David Pardo. Amount awarded: 112,400€.

• 2018–2023: “Multiscale Inversion of Porous Rock Physics using High-Performance Simulators: Bridging the Gap between Mathematics and Geophysics (MATHROCKS)” (Ref. No.: 777778). Marie Sklodowska-Curie grant, European Union’s Horizon 2020. Principal Investigator: David Pardo. Amount awarded: 765,000€.

Academic Service

Member of Organizing Committees:

• Minisymposium "Neural Networks for Solving PDEs" in the 13th Edition of the Biennial Congress on Numerical Methods (CMN), Gijón, Spain, July 2026.

• 6th Edition of the Minimum Residual & Least-Squares Finite Element Methods (MINRES) Workshop. Basque Center for Applied Mathematics, Bilbao, Spain, June 2024.

• Course and Working Group on “Parametric PDEs using Deep Learning”, Asturias, Spain, May 2023.

• Course and Working Group on “Solving Forward and Inverse Problems with Deep Learning”, Cantabria, Spain, May 2021.

• Course and Working Group on “Deep Learning for Solving Inverse Problems using TF2.0”, Asturias, Spain, June 2020.

Regular reviewer for journals and conferences:

• Computer Methods in Applied Mechanics and Engineering (CMAME), Elsevier.

• Computers and Mathematics with Applications (CAMWA), Elsevier.

• Neural Networks (NEUNET), Elsevier.

• Neurocomputing (NEUCOM), Elsevier.

• Journal of Computational Science (JOCS), Elsevier.

• Communications in Computational Physics (CiCP), Global Science Press.

• SEMA Journal, Springer.

• International Conference on Computational Science (ICCS).