Deep learning has shown remarkable success in various fields, and efforts continue to develop investment methodologies using deep learning in the financial sector. Despite numerous successes, the fact is that the revolutionary results seen in areas such as image processing and natural language processing have not been seen in finance. There are two reasons why deep learning has not led to disruptive change in finance. First, the scarcity of financial data leads to overfitting in deep learning models, so excellent backtesting results do not translate into actual outcomes. Second, there is a lack of methodological development for optimizing dynamic control models under general conditions. Therefore, I aim to overcome the first problem by artificially augmenting market data through an integration of Generative Adversarial Networks (GANs) and the Fama-French factor model, and to address the second problem by enabling optimal control even under complex conditions using policy-based reinforcement learning. The methods of this study have been shown to significantly outperform traditional linear financial factor models such as the CAPM and value-based approaches such as the HJB equation.

Quantum embedding is a fundamental prerequisite for applying quantum machine learning techniques to classical data, and has substantial impacts on performance outcomes. In this study, we present Neural Quantum Embedding (NQE), a method that efficiently optimizes quantum embedding beyond the limitations of positive and trace-preserving maps by leveraging classical deep learning techniques. NQE enhances the lower bound of the empirical risk, leading to substantial improvements in classification performance. Moreover, NQE improves robustness against noise. To validate the effectiveness of NQE, we conduct experiments on IBM quantum devices for image data classification, resulting in a remarkable accuracy enhancement. In addition, numerical analyses highlight that NQE simultaneously improves the trainability and generalization performance of quantum neural networks, as well as of the quantum kernel method.

The qualitative theory of dynamical systems mainly provides a mathematical framework for analyzing the long-time behavior of systems without necessarily finding solutions for the given ODEs. The theory of dynamical systems could be related to deep learning problems from various perspectives such as approximation, optimization, generalization, and explainability. In this talk, we first introduce the qualitative theory of dynamical systems. Then, we present numerical results as the application of the qualitative theory of dynamical systems to deep learning problems.