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Physics-informed machine learning


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.

Key points

  • 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.

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Fig. 1: Physics-inspired neural network architectures.
Fig. 2: Inferring the 3D flow over an espresso cup based using the Tomo-BOS imaging system and physics-informed neural networks (PINNs).
Fig. 3: Physics-informed filtering of in-vivo 4D-flow magnetic resonance imaging data of blood flow in a porcine descending aorta.
Fig. 4: Uncovering edge plasma dynamics.
Fig. 5: Transitions between metastable states.


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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).

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Authors are listed in alphabetical order. G.E.K. supervised the project. All authors contributed equally to writing the paper.

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Correspondence to George Em Karniadakis.

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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.

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Karniadakis, G.E., Kevrekidis, I.G., Lu, L. et al. Physics-informed machine learning. Nat Rev Phys 3, 422–440 (2021).

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