Mean-field equations, information inequalities and concentration bounds

博士项目专注于平均场方程的各种方面的分析及其在金融、物理和机器学习中的应用。我们发展了用于逼近McKeanVlasov/平均场方程的相互作用粒子系统的欧拉型随机近似的所谓浓度界的新结果。我们的目标是平均场型方程在各种应用中的新适定性。我们开发了一个信息论框架，用于量化和减少基于轨迹的预测中的误差，这些预测是从生成平均场型随机动力系统的不确定向量场获得的。这是出于在基于简化的数据驱动模型的多尺度系统中改进预测的必要性。这里，与真实动力学及其近似相关的两个概率度量之间的距离通过所谓的Phi-发散来定义。其目的是获得关于基于近似动力学的可观测数据估计中的不确定度的一般信息界。这一新的框架在基于现场的模型误差和由此产生的基于轨迹的预测的不确定性之间提供了系统的联系。我们试图更好地理解和发展瓦瑟斯坦梯度流理论及其对非耗散系统的推广。结果包括相关平均场方程及其近似粒子系统与潜在的随机动力系统的新的正则性陈述。

The PhD project focuses on the analysis of mean-field equations in its variety of aspects and applications in Finance, Physics and Machine Learning. We develop new results on so-called concentration bounds for stochastic approximations of Euler type for the interacting particle systems used to approximate McKeanVlasov/mean-field equations. We aim at new well-posedness of mean-field type equations in its varying applications. We develop an information-theoretic framework for quantification and mitigation of error in trajectory-based predictions which are obtained from uncertain vector fields generating the underlying stochastic dynamical system on mean-field type. This is motivated by the necessity to improve predictions in multi-scale systems based on simplified, data-driven models. Here, the distance between two probability measures associated with the true dynamics and its approximation is defined via so-called phi-divergences. The goal is toobtain general information bounds on the uncertainty in estimates of observables based on the approximate dynamics in terms of the phi-divergences. This new framework provides a systematic link between field-based model error and the resulting uncertainty in trajectory-based predictions. We seek to better understand and develop the theory of Wasserstein gradient flows and its generalization to non-dissipative systems. Results include new regularity statements for the associated mean-field equations and their approximating particle systems in association with underlying random dynamical systems.