Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD)⁵¹ and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables which spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky partical differential equation as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy, we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.

The committor function is a central object of study in understanding transitions between metastable states in complex systems. However, computing the committor function for realistic systems at low temperatures is a challenging task due to the curse of dimensionality and the scarcity of transition data. In this paper, we introduce a computational approach that overcomes these issues and achieves good performance on complex benchmark problems with rough energy landscapes or in high dimensions. The new approach combines deep learning, data sampling, and feature engineering techniques. This establishes an alternative practical method for studying rare transition events between metastable states in complex, high dimensional systems.

The development of automated high-throughput experimental platforms has enabled fast sampling of high-dimensional decision spaces. To reach target properties efficiently, these platforms are increasingly paired with intelligent experimental design. However, current optimizers show limitations in maintaining sufficient exploration/exploitation balance for problems dealing with multiple conflicting objectives and complex constraints. Here, we devise an Evolution-Guided Bayesian Optimization (EGBO) algorithm that integrates selection pressure in parallel with a q-Noisy Expected Hypervolume Improvement (qNEHVI) optimizer; this not only solves for the Pareto Front (PF) efficiently but also achieves better coverage of the PF while limiting sampling in the infeasible space. The algorithm is developed together with a custom self-driving lab for seed-mediated silver nanoparticle synthesis, targeting 3 objectives (1) optical properties, (2) fast reaction, and (3) minimal seed usage alongside complex constraints. We demonstrate that, with appropriate constraint handling, EGBO performance improves upon state-of-the-art qNEHVI. Furthermore, across various synthetic multi-objective problems, EGBO shows significative hypervolume improvement, revealing the synergy between selection pressure and the qNEHVI optimizer. We also demonstrate EGBO’s good coverage of the PF as well as comparatively better ability to propose feasible solutions. We thus propose EGBO as a general framework for efficiently solving constrained multi-objective problems in high-throughput experimentation platforms.

We propose a systematic method for learning stable and physically interpretable dynamical models using sampled trajectory data from physical processes based on a generalized Onsager principle. The learned dynamics are autonomous ordinary differential equations parametrized by neural networks that retain clear physical structure information, such as free energy, diffusion, conservative motion, and external forces. For high-dimensional problems with a low-dimensional slow manifold, an autoencoder with metric-preserving regularization is introduced to find the low-dimensional generalized coordinates on which we learn the generalized Onsager dynamics. Our method exhibits clear advantages over existing methods on benchmark problems for learning ordinary differential equations. We further apply this method to study Rayleigh-Bénard convection and learn Lorenz-like low-dimensional autonomous reduced-order models that capture both qualitative and quantitative properties of the underlying dynamics. This forms a general approach to building reduced-order models for forced-dissipative systems.

One of the most exciting applications of artificial intelligence is automated scientific discovery based on previously amassed data, coupled with restrictions provided by known physical principles, including symmetries and conservation laws. Such automated hypothesis creation and verification can assist scientists in studying complex phenomena, where traditional physical intuition may fail. Here we develop a platform based on a generalized Onsager principle to learn macroscopic dynamical descriptions of arbitrary stochastic dissipative systems directly from observations of their microscopic trajectories. Our method simultaneously constructs reduced thermodynamic coordinates and interprets the dynamics on these coordinates. We demonstrate its effectiveness by studying theoretically and validating experimentally the stretching of long polymer chains in an externally applied field. Specifically, we learn three interpretable thermodynamic coordinates and build a dynamical landscape of polymer stretching, including the identification of stable and transition states and the control of the stretching rate. Our general methodology can be used to address a wide range of scientific and technological applications.