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We prove rigorous bounds on the errors resulting from the approximation of the incompressible Navier-Stokes equations with (extended) physics informed neural networks.
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E, W., Han, J. & Jentzen, A. (2017), ‘Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations’, Communications in Mathematics and Statistics
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Arora, S., Ge, R., Neyshabur, B. & Zhang, Y. (2018), Stronger generalization bounds for deep nets via a compression approach, in
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Raissi, M. & Karniadakis, G. E. (2018), ‘Hidden physics models: Machine learning of nonlinear partial differential equations’, Journal of Computational Physics
2018
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2018
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Jin, X., Cai, S., Li, H. & Karniadakis, G. E. (2021), ‘NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations’, Journal of Computational Physics
2021
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Kovachki, N., Lanthaler, S. & Mishra, S. (2021), ‘On universal approximation and error bounds for Fourier Neural Operators’, Journal of Machine Learning Research
2021
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Kutyniok, G., Petersen, P., Raslan, M. & Schneider, R. (2021), ‘A theoretical analysis of deep neural networks and parametric PDEs’, Constructive Approximation
2021
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Lye, K. O., Mishra, S., Ray, D. & Chandrashekar, P. (2021), ‘Iterative surrogate model optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks’, Computer Methods in Applied Mechanics and Engineering
2021
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Pang, G., Lu, L. & Karniadakis, G. E. (2019), ‘fPINNs: Fractional physics-informed neural networks’, SIAM journal of Scientific computing
2019
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Raissi, M., Perdikaris, P. & Karniadakis, G. E. (2019), ‘Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations’, Journal of Computational Physics
2019
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Schwab, C. & Zech, J. (2019), ‘Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ’, Analysis and Applications
2019
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Jagtap, A. D., Kharazmi, E. & Karniadakis, G. E. (2020), ‘Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems’, Computer Methods in Applied Mechanics and Engineering
2020
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Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A. & Anandkumar, A. (2020), ‘Fourier neural operator for parametric partial differential equations’
2020
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Lye, K. O., Mishra, S. & Ray, D. (2020), ‘Deep learning observables in computational fluid dynamics’, Journal of Computational Physics
2020
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Mao, Z., Jagtap, A. D. & Karniadakis, G. E. (2020), ‘Physics-informed neural networks for high-speed flows.’, Computer Methods in Applied Mechanics and Engineering
2020
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2021
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Mishra, S. & Molinaro, R. (2021 a
2021
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Mishra, S. & Molinaro, R. (2021 b
2021
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Mishra, S. & Rusch, T. K. (2021), ‘Enhancing accuracy of deep learning algorithms by training with low-discrepancy sequences’, SIAM Journal on Numerical Analysis
2021
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Shukla, K., Jagtap, A. D., Blackshire, J. L., Sparkman, D. & Karniadakis, G. E. (2021), ‘A physics-informed neural network for quantifying the microstructural properties of polycrystalline nickel using ultrasound data: A promising approach for solving inverse problems’, IEEE Signal Processing Magazine
2021
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Shukla, K., Jagtap, A. D. & Karniadakis, G. E. (2021), ‘Parallel physics-informed neural networks via domain decomposition’, Journal of Computational Physics
2021
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Preprint, available from arXiv:2106:05384
Wang, S. & Perdikaris, P. (2021), Long-time integration of parametric evolution equations with physics-informed deeponets · 2021
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Yang, L., Meng, X. & Karniadakis, G. E. (2021), ‘B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data’, Journal of Computational Physics
2021
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2022
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Jagtap, A. D., Mitsotakis, D. & Karniadakis, G. E. (2022), ‘Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre–Green–Naghdi equations’, Ocean Engineering
2022
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Lanthaler, S., Mishra, S. & Karniadakis, G. E. (2022), ‘Error estimates for deeponets: A deep learning framework in infinite dimensions’, Transactions of Mathematics and Its Applications
2022
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Jagtap, A. D. & Karniadakis, G. E. (2020), ‘Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations’, Communications in Computational Physics
2041
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