Inferring solutions of differential equations using noisy multi-fidelity data (1607.04805v1)

Abstract: For more than two centuries, solutions of differential equations have been obtained either analytically or numerically based on typically well-behaved forcing and boundary conditions for well-posed problems. We are changing this paradigm in a fundamental way by establishing an interface between probabilistic machine learning and differential equations. We develop data-driven algorithms for general linear equations using Gaussian process priors tailored to the corresponding integro-differential operators. The only observables are scarce noisy multi-fidelity data for the forcing and solution that are not required to reside on the domain boundary. The resulting predictive posterior distributions quantify uncertainty and naturally lead to adaptive solution refinement via active learning. This general framework circumvents the tyranny of numerical discretization as well as the consistency and stability issues of time-integration, and is scalable to high-dimensions.

• The paper introduces a probabilistic framework that integrates Gaussian Process regression with integro-differential operators to infer solutions from noisy multi-fidelity data.
• It bypasses traditional discretization by employing adaptive active learning for high-dimensional and complex geometrical problems.
• Numerical experiments show reduced prediction errors and credible uncertainty estimates, demonstrating robustness across various differential equation challenges.

Overview of "Inferring Solutions of Differential Equations Using Noisy Multi-Fidelity Data"

This paper addresses a contemporary challenge in computational physics by introducing a data-driven approach to infer solutions of differential equations using multi-fidelity data contaminated with noise. The authors, Maziar Raissi, Paris Perdikaris, and George Em Karniadakis, propose a novel methodology that establishes a probabilistic framework for interfacing machine learning with the theory of differential equations.

Methodological Contributions

The central contribution of this work lies in integrating Gaussian Process (GP) regression with integro-differential operators to infer solutions from noisy, scattered data of varying fidelity. The proposed framework circumvents traditional numerical methods such as discretization, instead opting for a Bayesian inference approach that offers several advantages:

1. Scalability and Flexibility: The methodology scales effectively to high-dimensional spaces and can seamlessly adapt to complex geometrical domains.
2. Adaptive Learning: By leveraging probabilistic machine learning principles, the framework allows for adaptive refinement through active learning, which iteratively improves the solution by selecting optimal data points based on the uncertainty estimation.
3. Incorporation of Multi-Fidelity Data: It efficiently combines data across multiple levels of fidelity, making it suitable for models where high-fidelity data is expensive or scarce.

Numerical Results and Comparisons

The results presented in the paper demonstrate the capability of the proposed method to handle a variety of linear differential equation problems such as integro-differential equations in one dimension, time-dependent advection-diffusion-reaction equations, and high-dimensional Poisson equations. Noteworthy results include the accurate recovery of solutions without resorting to any spatial or temporal discretization. Furthermore, the probabilistic approach signifies its robustness by providing credible intervals that quantify the uncertainty inherent in predictions, a feature absent in classical deterministic methods.

In terms of performance, the multi-fidelity learning scheme effectively reduces prediction errors compared to single-fidelity approaches. For instance, when applied to a 10-dimensional Poisson equation, the algorithm autonomously identifies the effective dimensionality, highlighting its potential for handling high-dimensional problems without prior dimensionality reduction.

Theoretical and Practical Implications

The research presents significant theoretical advancements by proposing a seamless integration between linear differential equations and modern machine learning. Practically, this framework is applicable to various scientific domains requiring the solution of complex system equations, such as materials science, electromagnetism, and fluid dynamics. The potential for adaptively incorporating multi-source data enhances its utility in real-world scenarios where measurement noise and heterogeneous data fidelity are prevalent.

Future Directions

The paper opens several avenues for future research. Extending the method to nonlinear differential equations remains a challenge due to the complexity of defining appropriate priors. Furthermore, the exploration of more flexible deep kernel learning could augment the expressive capacity of GPs used in this framework. Incorporating non-Gaussian noise models would further enhance the method's applicability to real-world data.

In summary, the authors have meticulously developed a robust data-driven strategy, introducing a probabilistic numerical approach that promises to extend the frontiers of computational physics.