Physics-informed neural networks
In machine learning, physics-informed neural networks (PINNs), also referred to as theory-trained neural networks (TTNs), are a type of universal function approximator that can embed the knowledge of any physical laws that govern a given data-set in the learning process, and can be described by partial differential equations (PDEs). Low data availability for some biological and engineering problems limit the robustness of conventional machine learning models used for these applications. The prior knowledge of general physical laws acts in the training of neural networks (NNs) as a regularization agent that limits the space of admissible solutions, increasing the generalizability of the function approximation. This way, embedding this prior information into a neural network results in enhancing the information content of the available data, facilitating the learning algorithm to capture the right solution and to generalize well even with a low amount of training examples. Because they process continuous spatial and time coordinates and output continuous PDE solutions, they can be categorized as neural fields. == Function approximation == Most of the physical laws that govern the dynamics of a system can be described by partial differential equations. For example, the Navier–Stokes equations are a set of partial differential equations derived from the conservation laws (i.e., conservation of mass, momentum, and energy) that govern fluid mechanics. The solution of the Navier–Stokes equations with appropriate initial and boundary conditions allows the quantification of flow dynamics in a precisely defined geometry. However, these equations cannot be solved exactly and therefore numerical methods must be used (such as finite differences, finite elements and finite volumes). In this setting, these governing equations must be solved while accounting for prior assumptions, linearization, and adequate time and space discretization. Recently, solving the governing partial differential equations of physical phenomena using deep learning has emerged as a new field of scientific machine learning (SciML), leveraging the universal approximation theorem and high expressivity of neural networks. In general, deep neural networks could approximate any high-dimensional function given that sufficient training data are supplied. However, such networks do not consider the physical characteristics underlying the problem, and the level of approximation accuracy provided by them is still heavily dependent on careful specifications of the problem geometry as well as the initial and boundary conditions. Without this preliminary information, the solution is not unique and may lose physical correctness. To remedy this, Physics-Informed Neural Networks (PINNs) leverage governing physical equations in neural network training. Namely, PINNs are designed to be trained to satisfy the given training data as well as the imposed governing equations. In this fashion, a neural network can be guided with training datasets that do not necessarily need to be large or complete. An accurate solution of partial differential equations can potentially be found without knowing the boundary conditions. Therefore, with some knowledge about the physical characteristics of the problem and some form of training data (even sparse and incomplete), PINNs may be used for finding an optimal solution with high fidelity. PINNs can be applied to a wide range of problems in computational science, and are a pioneering technology leading to the development of new classes of numerical solvers for PDEs. PINNs can be thought of as a mesh-free alternative to traditional approaches (e.g., CFD for fluid dynamics), and new data-driven approaches for model inversion and system identification. Notably, a trained PINN network can be used to predict values on simulation grids of different resolutions without needing to be retrained. Additionally, the derivatives used in the partial differential equations can be computed using automatic differentiation (AD), which is assessed to be superior to numerical or symbolic differentiation. == Modeling and computation == A general nonlinear partial differential equation can be written as: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} where u ( t , x ) {\displaystyle u(t,x)} denotes the solution, N [ ⋅ ; λ ] {\displaystyle {\mathcal {N}}[\cdot ;\lambda ]} is a nonlinear operator parameterized by λ {\displaystyle \lambda } , and Ω {\displaystyle \Omega } is a subset of R D {\displaystyle \mathbb {R} ^{D}} . This general form of governing equations summarizes a wide range of problems in mathematical physics, such as conservative laws, diffusion process, advection-diffusion systems, and kinetic equations. Given noisy measurements of a generic dynamic system described by the equation above, PINNs can be designed to solve two classes of problems: data-driven solutions of partial differential equations data-driven discovery of partial differential equations === Data-driven solution of partial differential equations === The data-driven solution of PDE computes the hidden state u ( t , x ) {\displaystyle u(t,x)} of the system given boundary data and/or measurements z {\displaystyle z} , and fixed model parameters λ {\displaystyle \lambda } . We solve: u t + N [ u ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u]=0,\quad x\in \Omega ,\quad t\in [0,T]} . by defining the residual f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ] {\displaystyle f:=u_{t}+{\mathcal {N}}[u]} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network. This network can be differentiated using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} is the error between the PINN u ( t , x ) {\displaystyle u(t,x)} and the set of boundary conditions and measured data on the set of points Γ {\displaystyle \Gamma } where the boundary conditions and data are defined. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the mean-squared error of the residual function. This second term encourages the PINN to learn the structural information expressed by the PDE during the training process. This approach has been used to yield computationally efficient physics-informed surrogate models with applications in the forecasting of physical processes, model predictive control, multi-physics and multi-scale modeling, and simulation. It has been shown to converge to the solution of the PDE. === Data-driven discovery of partial differential equations === Given noisy and incomplete measurements z {\displaystyle z} of the state of the system, the data-driven discovery of PDEs results in computing the unknown state u ( t , x ) {\displaystyle u(t,x)} and learning model parameters λ {\displaystyle \lambda } that best describe the observed data: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} By defining f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ; λ ] = 0 {\displaystyle f:=u_{t}+{\mathcal {N}}[u;\lambda ]=0} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network, f ( t , x ) {\displaystyle f(t,x)} results in a PINN. This network can be derived using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} , together with the parameter λ {\displaystyle \lambda } of the differential operator can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} , with u {\displaystyle u} and z {\displaystyle z} state solutions and measurements at sparse location Γ {\displaystyle \Gamma } , respectively. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the residual function. This second term requires the structured information represented by the partial differential equations to be satisfied in the training process. This strategy allows for discovering dynamic models described by nonlinear PDEs assembling computationally efficient and fully differentiable surrogate models that may find application in predictive forecasting, control, and data assimilation. == Extensions and applications == === For piece-wise function approximation === PINNs are unable to approximate PDEs that have strong non-linearity or sharp gradients (such as those that commonly occur in practical fluid flow problems). Piecewise approximation has been an old practic
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