Sampling the phase space of molecular systems -and, more generally, of complex systems effectively modeled by stochastic differential equations (SDEs)- is a crucial modeling step in many fields, from protein folding to materials discovery. These problems are often multiscale in nature: they can be described in terms of low-dimensional effective free energy surfaces parametrized by a small number of "slow" reaction coordinates; the remaining "fast" degrees of freedom populate an equilibrium measure conditioned on the reaction coordinate values. Sampling procedures for such problems are used to estimate effective free energy differences as well as ensemble averages with respect to the conditional equilibrium distributions; these latter averages lead to closures for effective reduced dynamic models. Over the years, enhanced sampling techniques coupled with molecular simulation have been developed; they often use knowledge of the system order parameters in order to sample the corresponding conditional equilibrium distributions, and estimate ensemble averages of observables. An intriguing analogy arises with the field of machine learning (ML), where generative adversarial networks (GANs) can produce high-dimensional samples from low-dimensional probability distributions. This sample generation is what is called, in equation-free multiscale modeling, a "lifting process": it returns plausible (or realistic) high-dimensional space realizations of a model state, from information about its low-dimensional representation. In this work, we elaborate on this analogy, and we present an approach that couples physics-based simulations and biasing methods for sampling conditional distributions with ML-based conditional generative adversarial networks (cGANs) for the same task. The "coarse descriptors" on which we condition the fine scale realizations can either be known a priori or learned through nonlinear dimensionality reduction (here, using diffusion maps). We suggest that this may bring out the best features of both approaches: we demonstrate that a framework that couples cGANs with physics-based enhanced sampling techniques can improve multiscale SDE dynamical systems sampling, and even shows promise for systems of increasing complexity (here, simple molecules).

The identification of nonlinear dynamical systems from experimental time series and image series data became an important research theme in the early 1990s [25, 37, 36]. After lapsing for almost two decades, it is now experiencing a spectacular rebirth. A key element of the older work was the use of neural architectures [14, 37] (recurrent, convolutional, ResNet) motivated by traditional numerical analysis algorithms. Importantly, such architectures allow researchers to identify effective, coarse-grained, mean-field type evolution models from fine-scale (atomistic, molecular, agent-based) data [29, 5]. In this paper, we identify coarse-grained, effective stochastic differential equations (eSDE) for colloidal particle selfassembly based onfine-grained, Brownian dynamics simulations under the influence of electric fields [51, 11]. We demonstrate that the identified eSDE encodes accurately the physics of the Brownian Dynamic simulations and captures the dynamics of corresponding experimental data. Those experiments have previously been shown to quantitatively match to BD simulations at equilibrium in terms of time-averaged distribution functions [11, 18, 20]. Figure 1 shows a sample path of a latent space trajectory {t, φ(t)}

We introduce a data-driven approach to building reduced dynamical models through manifold learning; the reduced latent space is discovered using Diffusion Maps (a manifold learning technique) on time series data. A second round of Diffusion Maps on those latent coordinates allows the approximation of the reduced dynamical models. This second round enables mapping the latent space coordinates back to the full ambient space (what is called lifting); it also enables the approximation of full state functions of interest in terms of the reduced coordinates. In our work, we develop and test three different reduced numerical simulation methodologies, either through pre-tabulation in the latent space and integration on the fly or by going back and forth between the ambient space and the latent space. The data-driven latent space simulation results, based on the three different approaches, are validated through (a) the latent space observation of the full simulation through the Nystr\"om Extension formula, or through (b) lifting the reduced trajectory back to the full ambient space, via Latent Harmonics. Latent space modeling often involves additional regularization to favor certain properties of the space over others, and the mapping back to the ambient space is then constructed mostly independently from these properties; here, we use the same data-driven approach to construct the latent space and then map back to the ambient space.