The Fokker--Planck (FP) equation governing the evolution of the probability density function (PDF) is applicable to many disciplines, but it requires specification of the coefficients for each case...

Physics-informed neural networks (PINNs) were recently proposed in [18] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution, while a PDE-induced NN is coupled to the solution NN, and all differential operators are treated using automatic differentiation. Here, we first employ the standard PINN and a stochastic version, sPINN, to solve forward and inverse problems governed by a non-linear advection–diffusion–reaction (ADR) equation, assuming we have some sparse measurements of the concentration field at random or pre-selected locations. Subsequently, we attempt to optimise the hyper-parameters of sPINN by using the Bayesian optimisation method (meta-learning) and compare the results with the empirically selected hyper-parameters of sPINN. In particular, for the first part in solving the inverse deterministic ADR, we assume that we only have a few high-fidelity measurements, whereas the rest of the data is of lower fidelity. Hence, the PINN is trained using a composite multi-fidelity network, first introduced in [12], that learns the correlations between the multi-fidelity data and predicts the unknown values of diffusivity, transport velocity and two reaction constants as well as the concentration field. For the stochastic ADR, we employ a Karhunen–Loève (KL) expansion to represent the stochastic diffusivity, and arbitrary polynomial chaos (aPC) to represent the stochastic solution. Correspondingly, we design multiple NNs to represent the mean of the solution and learn each aPC mode separately, whereas we employ a separate NN to represent the mean of diffusivity and another NN to learn all modes of the KL expansion. For the inverse problem, in addition to stochastic diffusivity and concentration fields, we also aim to obtain the (unknown) deterministic values of transport velocity and reaction constants. The available data correspond to 7spatial points for the diffusivity and 20 space–time points for the solution, both sampled 2000 times. We obtain good accuracy for the deterministic parameters of the order of 1–2% and excellent accuracy for the mean and variance of the stochastic fields, better than three digits of accuracy. In the second part, we consider the previous stochastic inverse problem, and we use Bayesian optimisation to find five hyper-parameters of sPINN, namely the width, depth and learning rate of two NNs for learning the modes. We obtain much deeper and wider optimal NNs compared to the manual tuning, leading to even better accuracy, i.e., errors less than 1% for the deterministic values, and about an order of magnitude less for the stochastic fields.

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.

We investigate a quantitative bistable two-dimensional model (MeKS network) of gene expression dynamics describing the competence development in the Bacillus subtilis under the influence of Lévy as well as Brownian motions. To analyze the transitions between the vegetative and the competence regions therein, two dimensionless deterministic quantities, the mean first exit time (MFET) and the first escape probability, are determined from a microscopic perspective, as well as their averaged versions from a macroscopic perspective. The relative contribution factor $\lambda$ , the ratio of non-Gaussian and Gaussian noise strengths, is adopted to identify an optimum choice in these transitions. Additionally, we use a recent geometric concept, the stochastic basin of attraction (SBA), to exhibit a pictorial comprehension about the influence of the Lévy motion on the basin stability of the competence state. Our main results indicate that (i) the transitions between the vegetative and the competence regions can be induced by the noise intensities, the relative contribution factor $\lambda$ and the Lévy motion index $\alpha$ ; (ii) a higher noise intensity and a larger $\alpha$ with smaller jump magnitude make the MFET shorter, and the MFET as a function of $\lambda$ exhibits one maximum value, which is a signature of the noise-enhanced stability phenomenon for the vegetative state; (iii) a larger $\alpha$ makes the transition from the vegetative to the adjacent competence region to occur at the highest probability. The Lévy motion index ${\alpha }_0\approx 0.5$ (a larger jump magnitude with a lower frequency) is an ideal choice to implement the transition to the non-adjacent competence region; (iv) there is an expansion in SBA when $\alpha$ decreases.