Linearization Turns Neural Operators into Function-Valued Gaussian Processes
(2406.05072)Abstract
Modeling dynamical systems, e.g. in climate and engineering sciences, often necessitates solving partial differential equations. Neural operators are deep neural networks designed to learn nontrivial solution operators of such differential equations from data. As for all statistical models, the predictions of these models are imperfect and exhibit errors. Such errors are particularly difficult to spot in the complex nonlinear behaviour of dynamical systems. We introduce a new framework for approximate Bayesian uncertainty quantification in neural operators using function-valued Gaussian processes. Our approach can be interpreted as a probabilistic analogue of the concept of currying from functional programming and provides a practical yet theoretically sound way to apply the linearized Laplace approximation to neural operators. In a case study on Fourier neural operators, we show that, even for a discretized input, our method yields a Gaussian closure--a structured Gaussian process posterior capturing the uncertainty in the output function of the neural operator, which can be evaluated at an arbitrary set of points. The method adds minimal prediction overhead, can be applied post-hoc without retraining the neural operator, and scales to large models and datasets. We showcase the efficacy of our approach through applications to different types of partial differential equations.
Overview
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The paper introduces the Neural Operator Laplace Approximation (NOLA) to address the lack of uncertainty quantification in neural operators used for modeling dynamical systems governed by partial differential equations (PDEs).
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The proposed framework leverages Gaussian processes to provide a probabilistic interpretation of these neural operators' outputs, enhancing their reliability for scientific and engineering applications.
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Experimental evaluations on various dynamical systems demonstrate that NOLA provides more realistic sample paths and better-calibrated uncertainty estimates compared to baseline methods.
Linearization Turns Neural Operators into Function-Valued Gaussian Processes
Overview
The paper addresses the challenge of uncertainty quantification in neural operators used for modeling dynamical systems governed by partial differential equations (PDEs). Neural operators, and more specifically Fourier neural operators (FNOs), have shown significant promise in learning complex solution operators for such PDEs. While these models are widely used in weather forecasting, fluid dynamics, and more, their predictions often lack reliable uncertainty estimates, which are crucial for many scientific and engineering applications. This work introduces the Neural Operator Laplace Approximation (NOLA), a novel framework that leverages Gaussian processes (GPs) to provide a probabilistic interpretation of these neural operators' outputs.
Background and Motivation
Partial differential equations (PDEs) are fundamental in describing interactions in various physical, biological, and engineering systems. Neural operators provide a way to learn mappings from function spaces, usually representing the solution operators of these PDEs, making them discretization invariant and applicable to a wide range of scenarios. Among these, Fourier neural operators (FNOs) have become particularly popular. However, the complex behavior of dynamic systems poses a unique challenge, as prediction errors can be significant and hard to detect. This paper proposes using approximate Bayesian uncertainty quantification via function-valued Gaussian processes (fGPs), extending the neural operator's capabilities by linearizing them and applying the Laplace approximation method.
Methodological Contributions
The authors develop NOLA, which approximates neural operators as function-valued Gaussian processes that enable efficient uncertainty quantification without retraining the model. The primary methodological steps involve:
- Currying of Neural Operators: The neural operator, initially mapping between function spaces, is converted into an equivalent neural network that handles inputs as pairs of functions and points.
- Linearized Laplace Approximation (LLA): The network is linearized around the maximum a posteriori (MAP) estimate of the parameters, allowing for a second-order Taylor approximation to the negative log-posterior.
- Gaussian Currying: The linearized model's posterior parameters, interpreted as Gaussian random variables, are transformed into a function-valued Gaussian process. This process facilitates a structured Gaussian posterior over the operator's output, capturing the uncertainty in its output function.
Experimental Evaluation
The approach is evaluated on dynamical systems described by the Korteweg-de Vries (KdV), Kuramoto-Sivashinsky (KS), and Burgers’ equations. These models were trained on varying numbers of trajectories, testing how uncertainty quantification scales with different data sizes. The efficacy of NOLA was demonstrated through metrics such as Negative Log-Likelihood (NLL), Root Mean Square Error (RMSE), and the q-statistic (Q), where it outperformed baseline methods based on input and weight perturbations. Notably, NOLA provided more realistic sample paths and better-calibrated uncertainty estimates.
Implications and Future Work
NOLA offers both practical and theoretical advantages. Practically, it introduces minimal computational overhead and can be applied post-hoc to neural operators without retraining, making it scalable to large models and datasets. Theoretically, this framework bridges neural operators with Bayesian methods, integrating function-valued Gaussian processes to capture epistemic uncertainty effectively.
The implications of this work are substantial for fields relying on PDEs, such as climate science and engineering, where understanding the uncertainty in predictions can significantly impact decision-making processes. Future work may investigate extending the framework to multi-dimensional problems, improving the approximation methods, and exploring the interpolation error associated with constructing feature functions in Bayesian Fourier neural operators.
Conclusion
This paper advances the field of operator learning by providing a robust method for uncertainty quantification using function-valued Gaussian processes. The framework logically extends the linearized Laplace approximation to neural operators, formalizing the concept of Gaussian currying. This approach offers a structured probabilistic insight into neural operators' outputs, making it a valuable tool for scientists and engineers dealing with complex dynamical systems described by PDEs. Future developments in this area hold promise for further enhancing the accuracy and reliability of computational models in various application domains.
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