Mean-field equations, information inequalities and concentration bounds

该博士项目侧重于平均场方程在各个方面的分析及其在金融、物理和机器学习中的应用。对于用于近似McKeanVlasov/平均场方程的相互作用粒子系统，我们开发了所谓的欧拉型随机近似的浓度界的新结果。研究了平均场型方程在不同应用中的新适定性。我们开发了一个信息理论框架，用于量化和减轻基于轨迹的预测中的误差，这些预测来自不确定向量场，产生平均场类型的潜在随机动力系统。这是由于有必要改进基于简化的数据驱动模型的多尺度系统的预测。这里，与真实动力学相关的两个概率测度之间的距离及其近似值通过所谓的phi-divergence来定义。目标是在基于菲散度的近似动力学的基础上，获得可观测值估计的不确定性的一般信息边界。这个新的框架在基于油田的模型误差和基于轨迹的预测中产生的不确定性之间提供了系统的联系。我们寻求更好地理解和发展沃瑟斯坦梯度流理论及其推广到非耗散系统。结果包括新的规则语句的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关联。

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.