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Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
Maziar Raissi, Paris Perdikaris, George Em Karniadakis
TL;DR
The paper tackles nonlinear PDE inference when data are limited and conventional approaches may lack robustness. It develops physics-informed neural networks that encode governing laws through differentiable neural approximators and demonstrates data-driven solution methods across benchmark problems. The reported results include accurate PDE predictions, while the authors position the methods as complementary to mature classical numerical solvers.
Problem
Limited scientific data make nonlinear physical systems difficult to infer, while existing learning approaches can lack robustness and convergence guarantees.
Method
Physics-informed neural networks use deep networks and automatic differentiation to encode nonlinear PDE constraints while inferring latent solutions and constructing differentiable surrogate models.
Results
6.7 · 10^-4 relative L2-norm error was obtained for the Burgers equation, alongside accurate predictions for Schrödinger and nearly discontinuous Burgers solutions.
Takeaways & Limitations
The methods provide data-driven PDE solution algorithms and physics-informed surrogate models that encode underlying physical laws as prior information.
Takeaways & Limitations
Continuous-time models may require many collocation points in higher-dimensional problems, and the methods are not replacements for mature classical numerical solvers.
Abstract
from arXiv · showhide
We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this two part treatise, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct classes of algorithms, namely continuous time and discrete time models. The resulting neural networks form a new class of data-efficient universal function approximators that naturally encode any underlying physical laws as prior information. In this first part, we demonstrate how these networks can be used to infer solutions to partial differential equations, and obtain physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters.
1. Introduction
The paper addresses learning nonlinear physical systems when data are scarce and conventional machine-learning models lack robustness or convergence guarantees. It proposes physics-informed neural networks that incorporate physical laws as prior constraints for data-driven PDE solution and discovery.
- Limited data acquisition makes scientific inference difficult, while many modern machine-learning methods lack robustness and convergence guarantees.
- Physical laws and validated domain knowledge can regularize learning by constraining admissible solutions.Such prior information can reduce the space of physically plausible outputs, including unrealistic flow solutions.
- Gaussian-process approaches use operator-tailored representations to infer solutions, but their Bayesian prior assumptions may limit capacity and robustness for nonlinear problems.
- Physics-informed neural networks use deep networks and automatic differentiation to enforce symmetries, invariances, conservation principles, and nonlinear PDE constraints.The approach avoids committing to prior assumptions, linearization, or local time-stepping.
- The paper studies data-driven PDE solution and PDE discovery, focusing here on computing solutions with continuous- and discrete-time algorithms.The algorithms are evaluated across benchmark problems for general nonlinear PDEs with latent solutions and nonlinear operators.
2. Continuous Time Models
Continuous-time physics-informed neural networks approximate PDE solutions with deep networks whose automatically differentiated residuals are trained alongside initial and boundary data. Burgers’ and Schrödinger examples show accurate predictions from limited data, while collocation-point requirements become a bottleneck in higher dimensions.
- Method: The method defines f(t, x) as the PDE residual and approximates u(t, x) with a deep neural network, using automatic differentiation to construct f.The shared network parameters are learned by minimizing a mean squared error loss combining data and physics terms.
- Method: The training loss combines mean squared error on initial and boundary data with a residual term enforcing the governing equation at collocation points.The benchmarks use relatively few training points and optimize the loss with L-BFGS.
- Burgers’ equation: Using Nu = 100 initial and boundary points, the Burgers’ model predicts the latent solution with a relative L2 error of 6.7 · 10^-4.The 9-layer network has 20 neurons per hidden layer and uses 10,000 collocation points; training takes approximately 60 seconds on one NVIDIA Titan X GPU.
- Burgers’ equation: The Burgers’ prediction captures nonlinear behavior including a sharp internal layer around t = 0.4 without discretizing the spatio-temporal domain.The reported error is about two orders of magnitude lower than in the authors’ previous Gaussian-process approach.
- Limitations: Continuous-time models require many collocation points to enforce physics across the full domain, with the number increasing exponentially in higher dimensions.The paper introduces structured Runge–Kutta-based neural networks as an approach that circumvents this requirement.
3. Discrete Time Models
Discrete-time physics-informed neural networks combine Runge–Kutta time stepping with neural-network approximations to predict PDE solutions sequentially or across large time steps. In Burgers’ equation benchmarks, increasing Runge–Kutta stages and network capacity enabled accurate single-step predictions, while the approach also addressed nonlinear reaction–diffusion dynamics.
- Runge–Kutta formulation: Runge–Kutta discretization supports both implicit and explicit time-stepping schemes, determined by the coefficients a_ij, b_j, and c_j.The stage values represent the solution at intermediate times t_n + c_j∆t.
- Neural-network construction: The discrete-time model places a multi-output neural-network prior on the Runge–Kutta stage values and final solution, learning shared parameters by minimizing squared errors.The network takes x as input and outputs the stage values and u^(n+1)(x).
- Sequential prediction: Sequential prediction trains on data at t_n and domain boundaries, predicts u(t_(n+1),x), then reuses that prediction as initial data for later time steps.This procedure can advance the latent solution one step at a time.
- Burgers’ equation: 500 Runge–Kutta stages advanced Burgers’ equation from t = 0.1 to t = 0.9 in one step with relative L2 error 8.2·10^-4.The experiment used N_n = 250 initial data points and a network with four hidden layers of 50 neurons each.
- Burgers’ equation: 32 or more Runge–Kutta stages allowed very large time steps without sacrificing predictive accuracy or numerical stability, whereas low-stage schemes lost accuracy for ∆t > 0.2.The study varied q and ∆t with a fixed four-layer, 50-neuron architecture.
- Allen–Cahn equation: The discrete-time models were also applied to Allen–Cahn dynamics with periodic boundary conditions and nonlinear reaction–diffusion terms.The experiment used 200 initial data points and predicted t = 0.9 from t = 0.1 with ∆t = 0.8.
4. Summary and Discussion
Physics informed neural networks encode physical laws described by partial differential equations as universal function approximators for data-driven solution inference and surrogate modeling. The paper reports diverse computational-science results while emphasizing coexistence with classical numerical methods and identifying uncertainty quantification as future work.
- Contributions: Physics informed neural networks encode underlying physical laws described by partial differential equations while approximating solutions as universal function approximators.The methods target data-driven inference of solutions to general nonlinear partial differential equations.
- Results: 6.99 · 10−3 relative L2 error is reported for the Allen-Cahn equation prediction in Figure 4.The figure compares training data at t = 0.1 with prediction at t = 0.9.
- Applications: The methods support data-driven forecasting, model predictive control, and multi-physics or multi-scale modeling and simulation.
- Limitations and scope: The proposed methods are not replacements for mature classical PDE solvers, which can already meet practical robustness and computational-efficiency standards.The paper instead presents coexistence between classical methods such as Runge-Kutta schemes and deep neural networks.
- Future work: Quantifying uncertainty in neural-network predictions remains an unresolved question requiring further investigation.The paper notes that this was addressed in prior Gaussian-process work but is not captured by the present methodology.