Probabilities-Informed Machine Learning

As a natural evolution of traditional regression methods [3], ML models such as Support Vector Regression (SVR) [4] and Artificial Neural Networks (ANN) [5] have been developed to handle non-linear relationships and highdimensional datasets [6] with increasing accuracy and robustness. For instance, SVR has proven to be a robust regression tool because it can generalize well with limited data and capture nonlinear relationships using kernel functions [7]. Similarly, ANN, inspired by the neural architecture of the human brain, has become foundational to ML [5]. Typically, these methods use inputs (X) and outputs (Y) to construct surrogate models that aim to minimize the difference between the predicted and actual output values. These models have found applications across diverse fields, including engineering, medicine, and economics, demonstrating their versatility and potential [8], [9], [10]. In many real-world applications, additional prior information regarding the output model can be leveraged to enhance its accuracy and robustness [11] [12]. For instance, in physical systems, knowledge of the governing laws of physics has been successfully incorporated into ML by developing physics-informed neural networks (PINNs) [13], leading to improved efficiency and accuracy in prediction tasks [14]. In addition to physical laws, probabilistic information about the structure of the problem may also exist in practical scenarios [15]. Moreover, in many systems, the output variable is inherently probabilistic, necessitating models to approximate the probabilistic structure of the output [16].