Physics-Driven Learning for Inverse Problems in Quantum Chromodynamics

TL;DR

Techniques for physics-driven learning and inverse modelling of generative models to provide solutions for inverse problem in quantum chromodynamics are explored and the potential of physics-driven designs of generative models beyond QCD physics is emphasized.

Abstract

The integration of deep learning techniques and physics-driven designs is reforming the way we address inverse problems, in which accurate physical properties are extracted from complex data sets. This is particularly relevant for quantum chromodynamics (QCD), the theory of strong interactions, with its inherent limitations in observational data and demanding computational approaches. This perspective highlights advances and potential of physics-driven learning methods, focusing on predictions of physical quantities towards QCD physics, and drawing connections to machine learning(ML). It is shown that the fusion of ML and physics can lead to more efficient and reliable problem-solving strategies. Key ideas of ML, methodology of embedding physics priors, and generative models as inverse modelling of physical probability distributions are introduced. Specific applications cover first-principle lattice calculations, and QCD physics of hadrons, neutron stars, and heavy-ion collisions. These examples provide a structured and concise overview of how incorporating prior knowledge such as symmetry, continuity and equations into deep learning designs can address diverse inverse problems across different physical sciences.

Figures (5)

• Figure 1: Physics-driven deep learning. In a deep neural network model, the weights, $\{w\} = w_1, w_2, \dots, w_n$, connect the inputs, $\{x\} = x_1, x_2, \dots, x_n$, and the outputs, $y$, with summation $\Sigma$ and non-linear activation functions $f(u)$. In a single layer, the equation can be simplified as $y = f(\Sigma_{i=1}^n x_i w_i)$. The symmetries can be encoded within the weights, and other principles can be represented by different activation functions. Due to its differentiable properties, physics equations can be explicitly utilised in the back propagation (BP) algorithm. The physical data provides guidance for the outputs from deep models when computing the loss functions.
• Figure 2: Example of an L-CNN. Each layer preserves gauge equivariance on the lattice, such that convolutional, bilinear, activation or exponentiation layers can be combined for the use in inverse problems where the preservation of this symmetry is crucial. Contrary to conventional CNNs, they are robust to random and adversarial gauge transformations. Image from Favoni:2020reg.
• Figure 3: Automatic differential framework to reconstruct spectral functions from observations. Neural networks have outputs as a list representation of spectrum $\rho(\omega_i)$. The convolutional operation between $\rho(\omega_i)$ and kernel function $K(\omega, p)$, gives the predicted observations $D(p)$, as $D(p_j) = \sum_i^{N_\omega} \Delta \omega \rho(\omega_i) K(\omega_i, p_j)$. The difference between real observations and predicted one is utilised to compute the loss function, $\mathcal{L} = \sum_j^{N_p}(D(p_j) - \tilde{D}(p_j))^2$, for optimising the weights $\{\theta\}$ of neural networks, with gradient $\partial\mathcal{L}/\partial \theta$. The activation functions of the neural network can be set as Softplus to meet the positive definition principle.
• Figure 4: Conceptual figure for the ill-posed inverse problem from observational data to the theory. In the context of the neutron star physics, in the theoretical level, the mapping between the equation of state (EoS) and the mass-radius ($M$-$R$) relation is well-posed, but the inferred likely EoS has a probability distribution reflecting the quality and quantity of data.
• Figure 5: A schematic flow chart for QCD transition binary classification with CNNs using final particle spectra from HICs as input. (a) Quark gluon plasma (QGP). The hydrodynamic evolution of QGP with the crossover and first-order phase transition encoded, respectively. (b) Hadrons. The final observations collected by detectors are signals of different hadrons. (c) Convolutional neural networks (CNNs). CNNs are suitable for classification task in identifying the phase transition signals from hadron spectra which are preprocessed as image-type data.