A Deterministic Approximation to Neural SDEs.

Area of Science:

• Computational neuroscience
• Machine learning
• Stochastic processes

Background:

• Neural Stochastic Differential Equations (NSDEs) integrate neural networks with stochastic processes.
• While NSDEs excel at prediction, their uncertainty quantification capabilities remain largely unexplored.
• Existing methods for uncertainty estimation in NSDEs are computationally prohibitive.

Purpose of the Study:

• To address the computational challenges of uncertainty quantification in NSDEs.
• To develop a computationally efficient and accurate method for approximating the transition kernel of NSDEs.
• To improve both the uncertainty calibration and prediction accuracy of NSDE models.

Main Methods:

• A novel deterministic scheme is proposed to approximate the transition kernel of NSDEs.
• The method employs a bidimensional moment matching algorithm, operating vertically across neural network layers and horizontally across time.
• This approach utilizes effective approximations for computational efficiency.

Main Results:

• The proposed deterministic scheme accurately approximates the transition kernel, enabling reliable uncertainty estimation.
• The method achieves comparable uncertainty calibration to Monte Carlo sampling but at a significantly lower computational cost.
• Numerical stability during deterministic training enhances overall prediction accuracy.

Conclusions:

• The developed deterministic scheme offers a computationally affordable solution for uncertainty quantification in NSDEs.
• This method improves the reliability of uncertainty estimates and boosts prediction accuracy.
• It provides a practical approach for leveraging NSDEs in applications requiring robust uncertainty assessment.