A local squared Wasserstein-2 method for efficient reconstruction of models with uncertainty (2406.06825v1)

Abstract: In this paper, we propose a local squared Wasserstein-2 (W_2) method to solve the inverse problem of reconstructing models with uncertain latent variables or parameters. A key advantage of our approach is that it does not require prior information on the distribution of the latent variables or parameters in the underlying models. Instead, our method can efficiently reconstruct the distributions of the output associated with different inputs based on empirical distributions of observation data. We demonstrate the effectiveness of our proposed method across several uncertainty quantification (UQ) tasks, including linear regression with coefficient uncertainty, training neural networks with weight uncertainty, and reconstructing ordinary differential equations (ODEs) with a latent random variable.

• The paper introduces a novel local squared Wasserstein-2 loss for reconstructing models under uncertainty without prior distribution information.
• It leverages empirical distributions to match model outputs with observed data in tasks like linear regression, neural networks, and ODE reconstruction.
• Numerical experiments show minimal errors in predicted means and standard deviations, demonstrating robust performance in uncertainty quantification.

Analyzing the Local Squared Wasserstein-2 Method for Model Reconstruction under Uncertainty

The research paper addresses the inverse problem of reconstructing models involving uncertain latent variables or parameters through a novel method based on local squared Wasserstein-2 distance ($W_2$). The methodology is notable for its independence from prior knowledge of the distribution of the latent variables or parameters. This paper bridges a significant gap in uncertainty quantification (UQ) tasks by reconstructing the distributions of outputs associated with various inputs directly from empirical distribution data.

Methodology

The authors propose using the local squared $W_2$ distance as a loss function for model reconstruction. The $W_2$ distance measures the discrepancy between two probability distributions, which has previously been applied in various UQ tasks and generative adversarial networks. The practical advantage of the local squared $W_2$ method lies in its ability to efficiently evaluate empirical distributions from a finite number of observations.

Consider the following model incorporating uncertainty: $\bm{y}(\bm{x}; \omega) = \bm{f}(\bm{x}, \omega),$ where $\bm{x}$ is the input, $\omega$ is a latent random variable, and $\bm{f}(\cdot, \cdot)$ defines the functional relationship. The proposed method aims to reconstruct an approximate model $\hat{\bm{y}}(\bm{x}; \hat{\omega})$ such that the distribution of $\hat{\bm{y}}(\bm{x}; \hat{\omega})$ matches the empirical distribution of $\bm{y}(\bm{x}; \omega)$ for each $\bm{x}$.

The main contributions are:

1. Formulation and Analysis: The paper introduces a local squared $W_2$ loss function for reconstructing models, efficiently evaluated by empirical distributions from observed data.
2. Distinction from Bayesian Methods: Unlike traditional Bayesian approaches that require prior distributions, this method does not need prior knowledge of $\omega$, nor does it require an explicit form of $\bm{f}$.
3. Numerical Validation: The effectiveness of the method is showcased in different UQ tasks, including linear regression with coefficient uncertainty, training neural networks with weight uncertainty, and reconstructing ordinary differential equations (ODEs) with uncertain parameters.

Results and Discussion

Linear Regression with Coefficient Uncertainty

The paper applies the local squared $W_2$ method to a linear regression problem where the coefficients are normally distributed. The authors use 1000 training samples and test the performance of their method. The distributions of the predicted $\hat{\bm{y}}$ align well with the ground truth $\bm{y}$. The errors in reconstructed means and standard deviations are minimal, particularly when the neighborhood size $\delta$ is moderate, indicating that the method accurately reconstructs the model parameters with uncertainty.

Training a Neural Network with Weight Uncertainty

The method is further tested on a nonlinear model using a neural network with weight uncertainty. The local squared $W_2$ loss is compared with other commonly used UQ loss functions and a Bayesian neural network (BNN) method. The neural network trained using the local squared $W_2$ loss outperformed others, with the smallest errors in the predicted mean and standard deviation on the testing set.

The paper also discusses the impact of variations in the standard deviations of the latent parameters and input. Larger deviations led to poorer reconstruction, highlighting the method’s sensitivity to the distribution of training samples.

Concrete Compressive Strength Prediction

As a practical application, the proposed method is tested on the concrete compressive strength dataset to reconstruct the distribution based on selected continuous variables. The local squared $W_2$ method shows higher prediction accuracy in terms of both mean and standard deviation when compared to a neural network without weight uncertainty.

Reconstructing an ODE with Parameter Uncertainty

The method is applied to reconstruct an ODE with uncertain latent parameters. The paper demonstrates accurate reconstruction of the ODE model, showing the distributions of the reconstructed and ground truth trajectories align closely. The paper finds that larger values of initial condition variance and model parameter variance lead to higher reconstruction errors, emphasizing the importance of training data density.

Implications and Future Work

This research provides a versatile tool for reconstructing models under uncertainty without prior distribution knowledge. The local squared $W_2$ method's capacity to leverage empirical distributions from finite observations makes it broadly applicable in UQ tasks across various domains.

Future work could explore:

• Optimizing the size of the neighborhood $\delta$.
• Integrating the method with stochastic differential equation reconstruction.
• Applying the method to more complex UQ problems.

This work moves the field towards more flexible and data-efficient approaches to UQ, enabling better handling of uncertainties inherent in real-world models.