Differential equations, in particular evolutionary differential equations, are used to model, often with uncanny effectiveness, a vast number of natural and social phenomena. The study of dynamical behaviors of the underlying models, e.g. their stabilities, robustness, bifurcations, and complexities, has become truly essential to understand the phenomena being modeled. It is precisely because of the modeling effectiveness of evolutionary differential equations that their dynamical study is among the fastest growing and most active multidisciplinary research fields today. Not only is the existing theory of dynamics of evolutionary equations closely intertwined with many areas of mathematics, but also the concepts, methods, and paradigms, introduced in the study have become an indispensible component of a multitude of works of more applied nature, in disciplines including chemistry, physics, material science, mechanics, electrical engineering, as well as in the emerging applications of social sciences, not to mention the explosion during the last several decades of applications related to the biological and life sciences. As we push the boundaries of applicability of the existing theory, we routinely discover that new mathematical and computational tools are needed, often symbiotically. To witness, it is widely accepted that an important factor contributing to the explosive growth in studying dynamics of evolutionary differential equations has been the availability of sophisticated and powerful computational tools. Yet, huge systems, systems with multiple time scales, multi-physics models, stochastic differential equations, non-smooth evolutionary problems, still transcend our collective understanding, and exhibit inherent complexities which are requiring the development of sophisticated new modeling and numerical techniques, and whose validity will need to be validated by rigorous mathematics.

Given all these backgrounds and developments, there is a definite need to maintain a high level of scientific interaction among various areas of dynamics of evolutionary equations and meanwhile promote new mathematical research in the field for emerging applications. It is not an understatement to say that the enormous growth of the field of dynamics of differential equations, both in sophistication and in the number of successful modeling in the sciences, is representing a major challenge requiring knowledge of an increasingly diverse and sophisticated set of techniques, as well as keeping abreast of new applications. With all these in mind, the conference series was established with the general goals of presenting recent developments and state of the art knowledge and applications in the field, bringing together practitioners from different parts of the world and fostering collaborations, and providing advanced research training for graduate students and postdocs.

Besides carrying on the general goals of the series, this 4th conference will emphasize the stochastic modeling and data-driven computational aspects related to dynamics of evolutionary differential equations. Modeling using differential equations plays important roles in the prediction of many real-world problems. The common approach for such a prediction is to use either physical or mechanistic principles to establish the model then use available data and statistical tools to fit parameters of the model for future predictions. However, the increasing complexities involved in many emerging applications have demanded substantial innovations to the traditional modeling-prediction approach by overcoming challenges due to the existence of uncertainties and environmental noises, lacking of mechanisms, and the unavailability of useful data, etc. The inefficiency of almost all present Covid-19 predictions is precisely one of such challenging examples in which not only are new modeling approaches needed, but also the problem of lacking public health data on infections and vaccination – the actual driver of persistent infections, needs to be encountered in parameter fittings. For highly structurally complex systems or systems containing highly complex dynamics, it is well-known that computations or simulations using convectional numerical solvers are either very inaccurate or subject to enormous computational costs. A seemingly promising alternative in the computations of such systems is to incorporate dynamical or stochastic reductions, and to adopt data-driven schemes together with machine-learning techniques. This approach is still in its infancy and its further development requires not only rigorous theoretical analysis but also substantial verifications in real applications.