Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and high-dimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning has emerged as a promising alternative, but training deep neural networks requires big data, not always available for scientific problems. Instead, such networks can be trained from additional information obtained by enforcing the physical laws (for example, at random points in the continuous space-time domain). Such physics-informed learning integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks. Moreover, it may be possible to design specialized network architectures that automatically satisfy some of the physical invariants for better accuracy, faster training and improved generalization. Here, we review some of the prevailing trends in embedding physics into machine learning, present some of the current capabilities and limitations and discuss diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems.
Physics-informed machine learning integrates seamlessly data and mathematical physics models, even in partially understood, uncertain and high-dimensional contexts.
Kernel-based or neural network-based regression methods offer effective, simple and meshless implementations.
Physics-informed neural networks are effective and efficient for ill-posed and inverse problems, and combined with domain decomposition are scalable to large problems.
Operator regression, search for new intrinsic variables and representations, and equivariant neural network architectures with built-in physical constraints are promising areas of future research.
There is a need for developing new frameworks and standardized benchmarks as well as new mathematics for scalable, robust and rigorous next-generation physics-informed learning machines.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
Scientific Reports Open Access 20 January 2023
Preprocessing algorithms for the estimation of ordinary differential equation models with polynomial nonlinearities
Nonlinear Dynamics Open Access 16 January 2023
Scientific Reports Open Access 13 January 2023
Subscribe to Nature+
Get immediate online access to Nature and 55 other Nature journal
Subscribe to Journal
Get full journal access for 1 year
only $8.25 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
Hart, J. K. & Martinez, K. Environmental sensor networks: a revolution in the earth system science? Earth Sci. Rev. 78, 177–191 (2006).
Kurth, T. et al. Exascale deep learning for climate analytics (IEEE, 2018).
Reddy, D. S. & Prasad, P. R. C. Prediction of vegetation dynamics using NDVI time series data and LSTM. Model. Earth Syst. Environ. 4, 409–419 (2018).
Reichstein, M. et al. Deep learning and process understanding for data-driven earth system science. Nature 566, 195–204 (2019).
Alber, M. et al. Integrating machine learning and multiscale modeling — perspectives, challenges, and opportunities in the biological, biomedical, and behavioral sciences. NPJ Digit. Med. 2, 1–11 (2019).
Iten, R., Metger, T., Wilming, H., Del Rio, L. & Renner, R. Discovering physical concepts with neural networks. Phys. Rev. Lett. 124, 010508 (2020).
Raissi, M., Perdikaris, P. & Karniadakis, G. E. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019).
Schmidt, M. & Lipson, H. Distilling free-form natural laws from experimental data. Science 324, 81–85 (2009).
Brunton, S. L., Proctor, J. L. & Kutz, J. N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113, 3932–3937 (2016).
Jasak, H. et al. OpenFOAM: A C++ library for complex physics simulations. Int. Workshop Coupled Methods Numer. Dyn. 1000, 1–20 (2007).
Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995).
Jia, X. et al. Physics-guided machine learning for scientific discovery: an application in simulating lake temperature profiles. Preprint at arXiv https://arxiv.org/abs/2001.11086 (2020).
Lu, L., Jin, P., Pang, G., Zhang, Z. & Karniadakis, G. E. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat. Mach. Intell. 3, 218–229 (2021).
Kashefi, A., Rempe, D. & Guibas, L. J. A point-cloud deep learning framework for prediction of fluid flow fields on irregular geometries. Phys. Fluids 33, 027104 (2021).
Li, Z. et al. Fourier neural operator for parametric partial differential equations. in Int. Conf. Learn. Represent. (2021).
Yang, Y. & Perdikaris, P. Conditional deep surrogate models for stochastic, high-dimensional, and multi-fidelity systems. Comput. Mech. 64, 417–434 (2019).
LeCun, Y. & Bengio, Y. et al. Convolutional networks for images, speech, and time series. Handb. Brain Theory Neural Netw. 3361, 1995 (1995).
Mallat, S. Understanding deep convolutional networks. Phil. Trans. R. Soc. A 374, 20150203 (2016).
Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A. & Vandergheynst, P. Geometric deep learning: going beyond Euclidean data. IEEE Signal Process. Mag. 34, 18–42 (2017).
Cohen, T., Weiler, M., Kicanaoglu, B. & Welling, M. Gauge equivariant convolutional networks and the icosahedral CNN. Proc. Machine Learn. Res. 97, 1321–1330 (2019).
Owhadi, H. Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games. SIAM Rev. 59, 99–149 (2017).
Raissi, M., Perdikaris, P. & Karniadakis, G. E. Inferring solutions of differential equations using noisy multi-fidelity data. J. Comput. Phys. 335, 736–746 (2017).
Raissi, M., Perdikaris, P. & Karniadakis, G. E. Numerical Gaussian processes for time-dependent and nonlinear partial differential equations. SIAM J. Sci. Comput. 40, A172–A198 (2018).
Owhadi, H. Bayesian numerical homogenization. Multiscale Model. Simul. 13, 812–828 (2015).
Hamzi, B. & Owhadi, H. Learning dynamical systems from data: a simple cross-validation perspective, part I: parametric kernel flows. Physica D 421, 132817 (2021).
Reisert, M. & Burkhardt, H. Learning equivariant functions with matrix valued kernels. J. Mach. Learn. Res. 8, 385–408 (2007).
Owhadi, H. & Yoo, G. R. Kernel flows: from learning kernels from data into the abyss. J. Comput. Phys. 389, 22–47 (2019).
Winkens, J., Linmans, J., Veeling, B. S., Cohen, T. S. & Welling, M. Improved semantic segmentation for histopathology using rotation equivariant convolutional networks. in Conf. Med. Imaging Deep Learn. (2018).
Bruna, J. & Mallat, S. Invariant scattering convolution networks. IEEE Trans. Pattern Anal. Mach. Intell. 35, 1872–1886 (2013).
Kondor, R., Son, H. T., Pan, H., Anderson, B. & Trivedi, S. Covariant compositional networks for learning graphs. Preprint at arXiv https://arxiv.org/abs/1801.02144 (2018).
Tai, K. S., Bailis, P. & Valiant, G. Equivariant transformer networks. Proc. Int. Conf. Mach. Learn. 97, 6086–6095 (2019).
Pfau, D., Spencer, J. S., Matthews, A. G. & Foulkes, W. M. C. Ab initio solution of the many-electron Schrödinger equation with deep neural networks. Phys. Rev. Res. 2, 033429 (2020).
Pun, G. P., Batra, R., Ramprasad, R. & Mishin, Y. Physically informed artificial neural networks for atomistic modeling of materials. Nat. Commun. 10, 1–10 (2019).
Ling, J., Kurzawski, A. & Templeton, J. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166 (2016).
Jin, P., Zhang, Z., Zhu, A., Tang, Y. & Karniadakis, G. E. SympNets: intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems. Neural Netw. 132, 166–179 (2020).
Lusch, B., Kutz, J. N. & Brunton, S. L. Deep learning for universal linear embeddings of nonlinear dynamics. Nat. Commun. 9, 4950 (2018).
Lagaris, I. E., Likas, A. & Fotiadis, D. I. Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9, 987–1000 (1998).
Sheng, H. & Yang, C. PFNN: A penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries. J. Comput. Phys. 428, 110085 (2021).
McFall, K. S. & Mahan, J. R. Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions. IEEE Transac. Neural Netw. 20, 1221–1233 (2009).
Beidokhti, R. S. & Malek, A. Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques. J. Franklin Inst. 346, 898–913 (2009).
Lagari, P. L., Tsoukalas, L. H., Safarkhani, S. & Lagaris, I. E. Systematic construction of neural forms for solving partial differential equations inside rectangular domains, subject to initial, boundary and interface conditions. Int. J. Artif. Intell. Tools 29, 2050009 (2020).
Zhang, D., Guo, L. & Karniadakis, G. E. Learning in modal space: solving time-dependent stochastic PDEs using physics-informed neural networks. SIAM J. Sci. Comput. 42, A639–A665 (2020).
Dong, S. & Ni, N. A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks. J. Comput. Phys. 435, 110242 (2021).
Wang, B., Zhang, W. & Cai, W. Multi-scale deep neural network (MscaleDNN) methods for oscillatory stokes flows in complex domains. Commun. Comput. Phys. 28, 2139–2157 (2020).
Liu, Z., Cai, W. & Xu, Z. Q. J. Multi-scale deep neural network (MscaleDNN) for solving Poisson–Boltzmann equation in complex domains. Commun. Comput. Phys. 28, 1970–2001 (2020).
Mattheakis, M., Protopapas, P., Sondak, D., Di Giovanni, M. & Kaxiras, E. Physical symmetries embedded in neural networks. Preprint at arXiv https://arxiv.org/abs/1904.08991 (2019).
Cai, W., Li, X. & Liu, L. A phase shift deep neural network for high frequency approximation and wave problems. SIAM J. Sci. Comput. 42, A3285–A3312 (2020).
Darbon, J. & Meng, T. On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton-Jacobi partial differential equations. J. Comput. Phys. 425, 109907 (2021).
Sirignano, J. & Spiliopoulos, K. DGM: a deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375, 1339–1364 (2018).
Kissas, G. et al. Machine learning in cardiovascular flows modeling: predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks. Comput. Methods Appl. Mech. Eng. 358, 112623 (2020).
Zhu, Y., Zabaras, N., Koutsourelakis, P. S. & Perdikaris, P. Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. J. Comput. Phys. 394, 56–81 (2019).
Geneva, N. & Zabaras, N. Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks. J. Comput. Phys. 403, 109056 (2020).
Wu, J. L. et al. Enforcing statistical constraints in generative adversarial networks for modeling chaotic dynamical systems. J. Comput. Phys. 406, 109209 (2020).
Pfrommer, S., Halm, M. & Posa, M. Contactnets: learning of discontinuous contact dynamics with smooth, implicit representations. Preprint at arXiv https://arxiv.org/abs/2009.11193 (2020).
Erichson, N.B., Muehlebach, M. & Mahoney, M. W. Physics-informed autoencoders for Lyapunov-stable fluid flow prediction. Preprint at arXiv https://arxiv.org/abs/1905.10866 (2019).
Shah, V. et al. Encoding invariances in deep generative models. Preprint at arXiv https://arxiv.org/abs/1906.01626 (2019).
Geneva, N. & Zabaras, N. Transformers for modeling physical systems. Preprint at arXiv https://arxiv.org/abs/2010.03957 (2020).
Li, Z. et al. Multipole graph neural operator for parametric partial differential equations. in Adv. Neural Inf. Process. Syst. (2020).
Nelsen, N. H. & Stuart, A. M. The random feature model for input–output maps between Banach spaces. Preprint at arXiv https://arxiv.org/abs/2005.10224 (2020).
Cai, S., Wang, Z., Lu, L., Zaki, T. A. & Karniadakis, G. E. DeepM&Mnet: inferring the electroconvection multiphysics fields based on operator approximation by neural networks. J. Comput. Phys. 436, 110296 (2020).
Mao, Z., Lu, L., Marxen, O., Zaki, T. A. & Karniadakis, G. E. DeepM&Mnet for hypersonics: predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators. Preprint at arXiv https://arxiv.org/abs/2011.03349 (2020).
Meng, X. & Karniadakis, G. E. A composite neural network that learns from multi-fidelity data: application to function approximation and inverse PDE problems. J. Comput. Phys. 401, 109020 (2020).
Sirignano, J., MacArt, J. F. & Freund, J. B. DPM: a deep learning PDE augmentation method with application to large-eddy simulation. J. Comput. Phys. 423, 109811 (2020).
Lu, L. et al. Extraction of mechanical properties of materials through deep learning from instrumented indentation. Proc. Natl Acad. Sci. USA 117, 7052–7062 (2020).
Reyes, B., Howard, A. A., Perdikaris, P. & Tartakovsky, A. M. Learning unknown physics of non-Newtonian fluids. Preprint at arXiv https://arxiv.org/abs/2009.01658 (2020).
Wang, W. & Gómez-Bombarelli, R. Coarse-graining auto-encoders for molecular dynamics. NPJ Comput. Mater. 5, 1–9 (2019).
Rico-Martinez, R., Anderson, J. & Kevrekidis, I. Continuous-time nonlinear signal processing: a neural network based approach for gray box identification (IEEE, 1994).
Xu, K., Huang, D. Z. & Darve, E. Learning constitutive relations using symmetric positive definite neural networks. Preprint at arXiv https://arxiv.org/abs/2004.00265 (2020).
Huang, D. Z., Xu, K., Farhat, C. & Darve, E. Predictive modeling with learned constitutive laws from indirect observations. Preprint at arXiv https://arxiv.org/abs/1905.12530 (2019).
Xu, K., Tartakovsky, A. M., Burghardt, J. & Darve, E. Inverse modeling of viscoelasticity materials using physics constrained learning. Preprint at arXiv https://arxiv.org/abs/2005.04384 (2020).
Li, D., Xu, K., Harris, J. M. & Darve, E. Coupled time-lapse full-waveform inversion for subsurface flow problems using intrusive automatic differentiation. Water Resour. Res. 56, e2019WR027032 (2020).
Tartakovsky, A., Marrero, C. O., Perdikaris, P., Tartakovsky, G. & Barajas-Solano, D. Physics-informed deep neural networks for learning parameters and constitutive relationships in subsurface flow problems. Water Resour. Res. 56, e2019WR026731 (2020).
Xu, K. & Darve, E. Adversarial numerical analysis for inverse problems. Preprint at arXiv https://arxiv.org/abs/1910.06936 (2019).
Yang, Y., Bhouri, M. A. & Perdikaris, P. Bayesian differential programming for robust systems identification under uncertainty. Proc. R. Soc. A 476, 20200290 (2020).
Rackauckas, C. et al. Universal differential equations for scientific machine learning. Preprint at arXiv https://arxiv.org/abs/2001.04385 (2020).
Wang, S., Yu, X. & Perdikaris, P. When and why PINNs fail to train: a neural tangent kernel perspective. Preprint at arXiv https://arxiv.org/abs/2007.14527 (2020).
Wang, S., Wang, H. & Perdikaris, P. On the eigenvector bias of Fourier feature networks: from regression to solving multi-scale PDEs with physics-informed neural networks. Preprint at arXiv https://arxiv.org/abs/2012.10047 (2020).
Pang, G., Yang, L. & Karniadakis, G. E. Neural-net-induced Gaussian process regression for function approximation and PDE solution. J. Comput. Phys. 384, 270–288 (2019).
Wilson, A. G., Hu, Z., Salakhutdinov, R. & Xing, E. P. Deep kernel learning. Proc. Int. Conf. Artif. Intell. Stat. 51, 370–378 (2016).
Owhadi, H. Do ideas have shape? Plato’s theory of forms as the continuous limit of artificial neural networks. Preprint at arXiv https://arxiv.org/abs/2008.03920 (2020).
Owhadi, H. & Scovel, C. Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization: From a Game Theoretic Approach to Numerical Approximation and Algorithm Design (Cambridge Univ. Press, 2019).
Micchelli, C. A. & Rivlin, T. J. in Optimal Estimation in Approximation Theory (eds. Micchelli, C. A. & Rivlin, T. J.) 1–54 (Springer, 1977).
Sard, A. Linear Approximation (Mathematical Surveys 9, American Mathematical Society, 1963).
Larkin, F. Gaussian measure in Hilbert space and applications in numerical analysis. Rocky Mt. J. Math. 2, 379–421 (1972).
Sul’din, A. V. Wiener measure and its applications to approximation methods. I. Izv. Vyssh. Uchebn. Zaved. Mat. 3, 145–158 (1959).
Diaconis, P. Bayesian numerical analysis. Stat. Decision Theory Relat. Top. IV 1, 163–175 (1988).
Kimeldorf, G. S. & Wahba, G. A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Ann. Math. Stat. 41, 495–502 (1970).
Owhadi, H., Scovel, C. & Schäfer, F. Statistical numerical approximation. Not. Am. Math. Soc. 66, 1608–1617 (2019).
Tsai, Y. H. H., Bai, S., Yamada, M., Morency, L. P. & Salakhutdinov, R. Transformer dissection: a unified understanding of transformer’s attention via the lens of kernel. Preprint at arXiv https://arxiv.org/abs/1908.11775 (2019).
Kadri, H. et al. Operator-valued kernels for learning from functional response data. J. Mach. Learn. Res. 17, 1–54 (2016).
González-García, R., Rico-Martínez, R. & Kevrekidis, I. G. Identification of distributed parameter systems: a neural net based approach. Comput. Chem. Eng. 22, S965–S968 (1998).
Long, Z., Lu, Y., Ma, X. & Dong, B. PDE-Net: learning PDEs from data. Proc. Int. Conf. Mach. Learn. 80, 3208–3216 (2018).
He, J. & Xu, J. MgNet: a unified framework of multigrid and convolutional neural network. Sci. China Math. 62, 1331–1354 (2019).
He, K., Zhang, X., Ren, S. & Sun, J. Deep residual learning for image recognition (IEEE, 2016).
Rico-Martinez, R., Krischer, K., Kevrekidis, I., Kube, M. & Hudson, J. Discrete- vs. continuous-time nonlinear signal processing of Cu electrodissolution data. Chem. Eng. Commun. 118, 25–48 (1992).
Weinan, E. A proposal on machine learning via dynamical systems. Commun. Math. Stat. 5, 1–11 (2017).
Chen, T. Q., Rubanova, Y., Bettencourt, J. & Duvenaud, D. K. Neural ordinary differential equations. Adv. Neural Inf. Process. Syst. 31, 6571–6583 (2018).
Jia, J. & Benson, A. R. Neural jump stochastic differential equations. Adv. Neural Inf. Process. Syst. 32, 9847–9858 (2019).
Rico-Martinez, R., Kevrekidis, I. & Krischer, K. in Neural Networks for Chemical Engineers (ed. Bulsari, A. B.) 409–442 (Elsevier, 1995).
He, J., Li, L., Xu, J. & Zheng, C. ReLU deep neural networks and linear finite elements. J. Comput. Math. 38, 502–527 (2020).
Jagtap, A. D., Kharazmi, E. & Karniadakis, G. E. Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems. Comput. Methods Appl. Mech. Eng. 365, 113028 (2020).
Yang, L., Zhang, D. & Karniadakis, G. E. Physics-informed generative adversarial networks for stochastic differential equations. SIAM J. Sci. Comput. 42, A292–A317 (2020).
Pang, G., Lu, L. & Karniadakis, G. E. fPINNs: fractional physics-informed neural networks. SIAM J. Sci. Comput. 41, A2603–A2626 (2019).
Kharazmi, E., Zhang, Z. & Karniadakis, G. E. hp-VPINNs: variational physics-informed neural networks with domain decomposition. Comput. Methods Appl. Mech. Eng. 374, 113547 (2021).
Jagtap, A. D. & Karniadakis, G. E. Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Commun. Comput. Phys. 28, 2002–2041 (2020).
Raissi, M., Yazdani, A. & Karniadakis, G. E. Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations. Science 367, 1026–1030 (2020).
Yang, L., Meng, X. & Karniadakis, G. E. B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. J. Comput. Phys. 415, 109913 (2021).
Wang, S. & Perdikaris, P. Deep learning of free boundary and Stefan problems. J. Comput. Phys. 428, 109914 (2020).
Spigler, S. et al. A jamming transition from under-to over-parametrization affects generalization in deep learning. J. Phys. A 52, 474001 (2019).
Geiger, M. et al. Scaling description of generalization with number of parameters in deep learning. J. Stat. Mech. Theory Exp. 2020, 023401 (2020).
Belkin, M., Hsu, D., Ma, S. & Mandal, S. Reconciling modern machine-learning practice and the classical bias–variance trade-off. Proc. Natl Acad. Sci. USA 116, 15849–15854 (2019).
Geiger, M. et al. Jamming transition as a paradigm to understand the loss landscape of deep neural networks. Phys. Rev. E 100, 012115 (2019).
Mei, S., Montanari, A. & Nguyen, P. M. A mean field view of the landscape of two-layer neural networks. Proc. Natl Acad. Sci. USA 115, E7665–E7671 (2018).
Mehta, P. & Schwab, D. J. An exact mapping between the variational renormalization group and deep learning. Preprint at arXiv https://arxiv.org/abs/1410.3831 (2014).
Stoudenmire, E. & Schwab, D. J. Supervised learning with tensor networks. Adv. Neural Inf. Process. Syst. 29, 4799–4807 (2016).
Choromanska, A., Henaff, M., Mathieu, M., Arous, G. B. & LeCun, Y. The loss surfaces of multilayer networks. Proc. Artif. Intell. Stat. 38, 192–204 (2015).
Poole, B., Lahiri, S., Raghu, M., Sohl-Dickstein, J. & Ganguli, S. Exponential expressivity in deep neural networks through transient chaos. Adv. Neural Inf. Process. Syst. 29, 3360–3368 (2016).
Yang, G. & Schoenholz, S. Mean field residual networks: on the edge of chaos. Adv. Neural Inf. Process. Syst. 30, 7103–7114 (2017).
Poggio, T., Mhaskar, H., Rosasco, L., Miranda, B. & Liao, Q. Why and when can deep — but not shallow — networks avoid the curse of dimensionality: a review. Int. J. Autom. Comput. 14, 503–519 (2017).
Grohs, P., Hornung, F., Jentzen, A. & Von Wurstemberger, P. A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black–Scholes partial differential equations. Preprint at arXiv https://arxiv.org/abs/1809.02362 (2018).
Han, J., Jentzen, A. & Weinan, E. Solving high-dimensional partial differential equations using deep learning. Proc. Natl Acad. Sci. USA 115, 8505–8510 (2018).
Goodfellow, I. et al. Generative adversarial nets. Adv. Neural Inf. Process. Syst. 27, 2672–2680 (2014).
Brock, A., Donahue, J. & Simonyan, K. Large scale GAN training for high fidelity natural image synthesis. in Int. Conf. Learn. Represent. (2019).
Yu, L., Zhang, W., Wang, J. & Yu, Y. SeqGAN: sequence generative adversarial nets with policy gradient (AAAI Press, 2017).
Zhu, J.Y., Park, T., Isola, P. & Efros, A. A. Unpaired image-to-image translation using cycle-consistent adversarial networks (IEEE, 2017).
Yang, L., Daskalakis, C. & Karniadakis, G. E. Generative ensemble-regression: learning particle dynamics from observations of ensembles with physics-informed deep generative models. Preprint at arXiv https://arxiv.org/abs/2008.01915 (2020).
Lanthaler, S., Mishra, S. & Karniadakis, G. E. Error estimates for DeepONets: a deep learning framework in infinite dimensions. Preprint at arXiv https://arxiv.org/abs/2102.09618 (2021).
Deng, B., Shin, Y., Lu, L., Zhang, Z. & Karniadakis, G. E. Convergence rate of DeepONets for learning operators arising from advection–diffusion equations. Preprint at arXiv https://arxiv.org/abs/2102.10621 (2021).
Xiu, D. & Karniadakis, G. E. The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002).
Marzouk, Y. M., Najm, H. N. & Rahn, L. A. Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comput. Phys. 224, 560–586 (2007).
Stuart, A. M. Inverse problems: a Bayesian perspective. Acta Numerica 19, 451 (2010).
Tripathy, R. K. & Bilionis, I. Deep UQ: learning deep neural network surrogate models for high dimensional uncertainty quantification. J. Comput. Phys. 375, 565–588 (2018).
Karumuri, S., Tripathy, R., Bilionis, I. & Panchal, J. Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks. J. Comput. Phys. 404, 109120 (2020).
Yang, Y. & Perdikaris, P. Adversarial uncertainty quantification in physics-informed neural networks. J. Comput. Phys. 394, 136–152 (2019).
Raissi, M., Perdikaris, P. & Karniadakis, G. E. Machine learning of linear differential equations using Gaussian processes. J. Comput. Phys. 348, 683–693 (2017).
& Fan, D. et al. A robotic intelligent towing tank for learning complex fluid-structure dynamics. Sci. Robotics 4, eaay5063 (2019).
Winovich, N., Ramani, K. & Lin, G. ConvPDE-UQ: convolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains. J. Comput. Phys. 394, 263–279 (2019).
Zhang, D., Lu, L., Guo, L. & Karniadakis, G. E. Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. J. Comput. Phys. 397, 108850 (2019).
Gal, Y. & Ghahramani, Z. Dropout as a Bayesian approximation: representing model uncertainty in deep learning. Proc. Int. Conf. Mach. Learn. 48, 1050–1059 (2016).
Cai, S. et al. Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks. J. Fluid Mech. 915 (2021).
Mathews, A., Francisquez, M., Hughes, J. & Hatch, D. Uncovering edge plasma dynamics via deep learning from partial observations. Preprint at arXiv https://arxiv.org/abs/2009.05005 (2020).
Rotskoff, G. M. & Vanden-Eijnden, E. Learning with rare data: using active importance sampling to optimize objectives dominated by rare events. Preprint at arXiv https://arxiv.org/abs/2008.06334 (2020).
Patel, R. G. et al. Thermodynamically consistent physics-informed neural networks for hyperbolic systems. Preprint at https://arxiv.org/abs/2012.05343 (2020).
Shukla, K., Di Leoni, P. C., Blackshire, J., Sparkman, D. & Karniadakis, G. E. Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks. J. Nondestruct. Eval. 39, 1–20 (2020).
Behler, J. & Parrinello, M. Generalized neural-network representation of high-dimensional potential-energy surfaces. Phys. Rev. Lett. 98, 146401 (2007).
Zhang, L., Han, J., Wang, H., Car, R. & Weinan, E. Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics. Phys. Rev. Lett. 120, 143001 (2018).
Jia, W. et al. Pushing the limit of molecular dynamics with ab initio accuracy to 100 million atoms with machine learning. Preprint at arXiv https://arxiv.org/abs/2005.00223 (2020).
Nakata, A. et al. Large scale and linear scaling DFT with the CONQUEST code. J. Chem. Phys. 152, 164112 (2020).
Zhu, W., Xu, K., Darve, E. & Beroza, G. C. A general approach to seismic inversion with automatic differentiation. Preprint at arXiv https://arxiv.org/abs/2003.06027 (2020).
Abadi, M. et al. Tensorflow: a system for large-scale machine learning. Proc. OSDI 16, 265–283 (2016).
Paszke, A. et al. PyTorch: an imperative style, high-performance deep learning library. Adv. Neural Inf. Process. Syst. 32, 8026–8037 (2019).
Chollet, F. et al. Keras — Deep learning library. Keras https://keras.io (2015).
Frostig, R., Johnson, M. J. & Leary, C. Compiling machine learning programs via high-level tracing. in Syst. Mach. Learn. (2018).
Lu, L., Meng, X., Mao, Z. & Karniadakis, G. E. DeepXDE: a deep learning library for solving differential equations. SIAM Rev. 63, 208–228 (2021).
Hennigh, O. et al. NVIDIA SimNet: an AI-accelerated multi-physics simulation framework. Preprint at arXiv https://arxiv.org/abs/2012.07938 (2020).
Koryagin, A., Khudorozkov, R. & Tsimfer, S. PyDEns: a Python framework for solving differential equations with neural networks. Preprint at arXiv https://arxiv.org/abs/1909.11544 (2019).
Chen, F. et al. NeuroDiffEq: A python package for solving differential equations with neural networks. J. Open Source Softw. 5, 1931 (2020).
Rackauckas, C. & Nie, Q. DifferentialEquations.jl — a performant and feature-rich ecosystem for solving differential equations in Julia. J. Open Res. Softw. 5, 15 (2017).
Haghighat, E. & Juanes, R. SciANN: a Keras/TensorFlow wrapper for scientific computations and physics-informed deep learning using artificial neural networks. Comput. Meth. Appl. Mech. Eng. 373, 113552 (2020).
Xu, K. & Darve, E. ADCME: Learning spatially-varying physical fields using deep neural networks. Preprint at arXiv https://arxiv.org/abs/2011.11955 (2020).
Gardner, J. R., Pleiss, G., Bindel, D., Weinberger, K. Q. & Wilson, A. G. Gpytorch: blackbox matrix–matrix Gaussian process inference with GPU acceleration. Adv. Neural Inf. Process. Syst. 31, 7587–7597 (2018).
Novak, R. et al. Neural Tangents: fast and easy infinite neural networks in Python. in Conf. Neural Inform. Process. Syst. (2020).
Xu, K. & Darve, E. Physics constrained learning for data-driven inverse modeling from sparse observations. Preprint at arXiv https://arxiv.org/abs/2002.10521 (2020).
Xu, K. & Darve, E. The neural network approach to inverse problems in differential equations. Preprint at arXiv https://arxiv.org/abs/1901.07758 (2019).
Xu, K., Zhu, W. & Darve, E. Distributed machine learning for computational engineering using MPI. Preprint at arXiv https://arxiv.org/abs/2011.01349 (2020).
Elsken, T., Metzen, J. H. & Hutter, F. Neural architecture search: a survey. J. Mach. Learn. Res. 20, 1–21 (2019).
He, X., Zhao, K. & Chu, X. AutoML: a survey of the state-of-the-art. Knowl. Based Syst. 212, 106622 (2021).
Hospedales, T., Antoniou, A., Micaelli, P. & Storkey, A. Meta-learning in neural networks: a survey. Preprint at arXiv https://arxiv.org/abs/2004.05439 (2020).
Xu, Z.-Q. J., Zhang, Y., Luo, T., Xiao, Y. & Ma, Z. Frequency principle: Fourier analysis sheds light on deep neural networks. Commun. Comput. Phys. 28, 1746–1767 (2020).
Rahaman, N. et al. On the spectral bias of neural networks. Proc. Int. Conf. Mach. Learn. 97, 5301–5310 (2019).
Ronen, B., Jacobs, D., Kasten, Y. & Kritchman, S. The convergence rate of neural networks for learned functions of different frequencies. Adv. Neural Inf. Process. Syst. 32, 4761–4771 (2019).
Cao, Y., Fang, Z., Wu, Y., Zhou, D. X. & Gu, Q. Towards understanding the spectral bias of deep learning. Preprint at arXiv https://arxiv.org/abs/1912.01198 (2019).
Wang, S., Teng, Y. & Perdikaris, P. Understanding and mitigating gradient pathologies in physics-informed neural networks. Preprint at arXiv https://arxiv.org/abs/2001.04536 (2020).
Tancik, M. et al. Fourier features let networks learn high frequency functions in low dimensional domains. Adv. Neural Inf. Process. Syst. 33 (2020).
Cai, W. & Xu, Z. Q. J. Multi-scale deep neural networks for solving high dimensional PDEs. Preprint at arXiv https://arxiv.org/abs/1910.11710 (2019).
Arbabi, H., Bunder, J. E., Samaey, G., Roberts, A. J. & Kevrekidis, I. G. Linking machine learning with multiscale numerics: data-driven discovery of homogenized equations. JOM 72, 4444–4457 (2020).
Owhadi, H. & Zhang, L. Metric-based upscaling. Commun. Pure Appl. Math. 60, 675–723 (2007).
Blum, A. L. & Rivest, R. L. Training a 3-node neural network is NP-complete. Neural Netw. 5, 117–127 (1992).
Lee, J. D., Simchowitz, M., Jordan, M. I. & Recht, B. Gradient descent only converges to minimizers. Annu. Conf. Learn. Theory 49, 1246–1257 (2016).
Jagtap, A. D., Kawaguchi, K. & Em Karniadakis, G. Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks. Proc. R. Soc. A 476, 20200334 (2020).
Wight, C. L. & Zhao, J. Solving Allen–Cahn and Cahn–Hilliard equations using the adaptive physics informed neural networks. Preprint at arXiv https://arXiv.org/abs/2007.04542 (2020).
Goswami, S., Anitescu, C., Chakraborty, S. & Rabczuk, T. Transfer learning enhanced physics informed neural network for phase-field modeling of fracture. Theor. Appl. Fract. Mech. 106, 102447 (2020).
Betancourt, M. A geometric theory of higher-order automatic differentiation. Preprint at arXiv https://arxiv.org/abs/1812.11592 (2018).
Bettencourt, J., Johnson, M. J. & Duvenaud, D. Taylor-mode automatic differentiation for higher-order derivatives in JAX. in Conf. Neural Inform. Process. Syst. (2019).
Newman, D, Hettich, S., Blake, C. & Merz, C. UCI repository of machine learning databases. ICS http://www.ics.uci.edu/~mlearn/MLRepository.html (1998).
Bianco, S., Cadene, R., Celona, L. & Napoletano, P. Benchmark analysis of representative deep neural network architectures. IEEE Access 6, 64270–64277 (2018).
Vlachas, P. R. et al. Backpropagation algorithms and reservoir computing in recurrent neural networks for the forecasting of complex spatiotemporal dynamics. Neural Networks (2020).
Shin, Y., Darbon, J. & Karniadakis, G. E. On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Commun. Comput. Phys. 28, 2042–2074 (2020).
Mishra, S. & Molinaro, R. Estimates on the generalization error of physics informed neural networks (PINNs) for approximating PDEs. Preprint at arXiv https://arxiv.org/abs/2006.16144 (2020).
Mishra, S. & Molinaro, R. Estimates on the generalization error of physics informed neural networks (PINNs) for approximating PDEs II: a class of inverse problems. Preprint at arXiv https://arxiv.org/abs/2007.01138 (2020).
Shin, Y., Zhang, Z. & Karniadakis, G.E. Error estimates of residual minimization using neural networks for linear PDEs. Preprint at arXiv https://arxiv.org/abs/2010.08019 (2020).
Kharazmi, E., Zhang, Z. & Karniadakis, G. Variational physics-informed neural networks for solving partial differential equations. Preprint at arXiv https://arxiv.org/abs/1912.00873 (2019).
Jo, H., Son, H., Hwang, H. Y. & Kim, E. Deep neural network approach to forward-inverse problems. Netw. Heterog. Media 15, 247–259 (2020).
Guo, M. & Haghighat, E. An energy-based error bound of physics-informed neural network solutions in elasticity. Preprint at arXiv https://arxiv.org/abs/2010.09088 (2020).
Lee, J. Y., Jang, J. W. & Hwang, H. J. The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the deep neural network approach. Preprint at arXiv https://arxiv.org/abs/2009.13280 (2020).
Kingma, D. P. & Ba, J. Adam: a method for stochastic optimization. in Int. Conf. Learn. Represent. (2015).
Luo, T. & Yang, H. Two-layer neural networks for partial differential equations: optimization and generalization theory. Preprint at arXiv https://arxiv.org/abs/2006.15733 (2020).
Jacot, A., Gabriel, F. & Hongler, C. Neural tangent kernel: convergence and generalization in neural networks. Adv. Neural Inf. Process. Syst. 31, 8571–8580 (2018).
Alnæs, M. et al. The FEniCS project version 1.5. Arch. Numer. Softw. 3, 9–23 (2015).
Kemeth, F. P. et al. An emergent space for distributed data with hidden internal order through manifold learning. IEEE Access 6, 77402–77413 (2018).
Kemeth, F. P. et al. Learning emergent PDEs in a learned emergent space. Preprint at arXiv https://arxiv.org/abs/2012.12738 (2020).
Defense Advanced Research Projects Agency. DARPA shredder challenge rules. DARPA https://web.archive.org/web/20130221190250/http://archive.darpa.mil/shredderchallenge/Rules.aspx (2011).
Rovelli, C. Forget time. Found. Phys. 41, 1475 (2011).
Hy, T. S., Trivedi, S., Pan, H., Anderson, B. M. & Kondor, R. Predicting molecular properties with covariant compositional networks. J. Chem. Phys. 148, 241745 (2018).
Hachmann, J. et al. The Harvard clean energy project: large-scale computational screening and design of organic photovoltaics on the world community grid. J. Phys. Chem. Lett. 2, 2241–2251 (2011).
Byrd, R. H., Lu, P., Nocedal, J. & Zhu, C. A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16, 1190–1208 (1995).
We thank H. Owhadi (Caltech) for his insightful comments on the connections between NNs and kernel methods. G.E.K. acknowledges support from the DOE PhILMs project (no. DE-SC0019453) and OSD/AFOSR MURI grant FA9550-20-1-0358. I.G.K. acknowledges support from DARPA (PAI and ATLAS programmes) as well as an AFOSR MURI grant through UCSB. P.P. acknowledges support from the DARPA PAI programme (grant HR00111890034), the US Department of Energy (grant DE-SC0019116), the Air Force Office of Scientific Research (grant FA9550-20-1-0060), and DOE-ARPA (grant 1256545).
The authors declare no competing interests.
Peer review information
Nature Reviews Physics thanks the anonymous reviewers for their contribution to the peer review of this work.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Neural Tangents: https://github.com/google/neural-tangents
- Multi-fidelity data
Data of variable accuracy.
- Lax–Oleinik formula
A representation formula for the solution of the Hamilton–Jacobi equation.
- Deep Galerkin method
A physics-informed neural network-like method with random sampling.
- Lyapunov stability
Characterization of the robustness of dynamic behaviour to small perturbations, in the neighbourhood of an equilibrium.
- Gappy data
Sets with regions of missing data.
- ReLU activation function
Rectified linear unit.
- Double-descent phenomenon
Increasing model capacity beyond the point of interpolation resulting in improved performance.
- Restricted Boltzmann machines
Generative stochastic artificial neural networks that can learn a probability distribution over their set of inputs.
- Aleatoric uncertainty
Uncertainty due to the inherent randomness of data.
- Epistemic uncertainty
Uncertainty due to limited data and knowledge.
- Arbitrary polynomial chaos
A type of generalized polynomial chaos with measures defined by data.
- Boussinesq approximation
An approximation used in gravity-driven flows, which ignores density differences except in the gravity term.
- Committor function
A function used to study transitions between metastable states in stochastic systems.
- Allen–Cahn type system
A type of system with both reaction and diffusion.
Dual refinement of the mesh by increasing either the number of subdomains or the approximations degree.
- Hölder regularization
A regularization term associated with Hölder constants of differential equations that controls the derivatives of neural networks.
- Rademacher complexity
A quantity that measures richness of a class of real-valued functions with respect to a probability distribution.
- Koopman model
Linear model of a (nonlinear) dynamical system obtained via a Koopman operator theory.
- Nesterov iterations
Iterations of an algorithm for the numerical computation of equilibria.
A nonlinear dimensionality reduction technique for embedding intrinsically low-dimensional data from high-dimensional representations to lower-dimensional spaces.
t-distributed stochastic neighbour embedding. A nonlinear dimensionality reduction technique for embedding intrinsically low-dimensional data from high-dimensional representations to lower-dimensional spaces.
- Diffusion maps
A nonlinear dimensionality reduction technique for embedding intrinsically low-dimensional data from high-dimensional representations to lower-dimensional spaces.
About this article
Cite this article
Karniadakis, G.E., Kevrekidis, I.G., Lu, L. et al. Physics-informed machine learning. Nat Rev Phys 3, 422–440 (2021). https://doi.org/10.1038/s42254-021-00314-5
This article is cited by
Scientific Reports (2023)
Nature Reviews Genetics (2023)
Scientific Reports (2023)
Scientific Reports (2023)
Recent developments in modeling, imaging, and monitoring of cardiovascular diseases using machine learning
Biophysical Reviews (2023)