Organizations: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076, Singapore · Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, 117585, Singapore · Institute for Functional Intelligent Materials, National University of Singapore, 4 Science Drive 2, 117544, Singapore · Fujitsu Research of Europe, 3–9 Albert Street, Slough SL1 2BE, United Kingdom
Using machine learning, we identify a second transition in polymer chain stretching driven by planar elongational flow, beyond the well-known coil-stretch transition. We achieve this by developing a conditional extension of Universal and Identifiable Dissipative Dynamics (UIDD) that jointly learns reduced stochastic dynamics from microscopic trajectories across flow strengths and yields the corresponding stationary nonequilibrium free energy landscapes. As the flow strength increases, the evolving landscape exhibits the expected coil-stretch transition, and then a previously unrecognized dynamical transition. In the latter, the stretched states persist, but the distribution along a folding coordinate becomes bimodal when conditioned on small end-to-end separation. Large-deviation analysis of a simplified analytical chain model reproduces both transitions and shows that they are phase transitions in the long-chain limit. Single-molecule DNA experiments provide qualitative support for the dynamical transition through the predicted reversal of the folding response.
Figures & tables
Figure 1: Overview of the polymer system and transition analysis. ( a ) A single polymer chain is represented by a bead-rod model, subjected to planar elongational flow, with chain extension measured along the elongational axis. ( b ) Plot of chain extension as a function of time across varying flow strengths. ( c ) Microscopic trajectories are reduced by PCA and fitted with conditional UIDD to identify phase transitions; large-deviation theory independently supports both transitions, while single-molecule DNA experiments provide qualitative support for the second transition.
Figure 2: Free energy landscapes and transition diagrams. ( a-d ) Learned marginal free energies F2(Z2) and sections F23(0,Z3) at the indicated Wi . Insets show the corresponding landscapes F23(Z2,Z3) . ( e,f ) Near-minimizing sets of the learned F2 ( η=0.6 ) and F23(0,⋅) ( η=1.0 ), respectively, defined by {q:F(q;Wi)−minq′F(q′;Wi)≤η} . Color intensity indicates the number of the five independently trained models satisfying this condition. For each model, the transition point Wit is determined by the first crossing of F(0;Wi)−minqF(q;Wi) above η . Orange dashed lines and bands indicate the mean and range of Wit across models, respectively, with sensitivities shown in the insets. ( g-j ) Free energy profiles per bond of the analytical model, F2,n(z2) and F23,n(0,z3) . ( k,l ) Minimizing sets of F2,∞(z2) and F23,∞(0,z3) , respectively. Beyond the second transition, the gray region in ( l ) denotes the minimizing interval, with diamonds marking its endpoints.
Figure 3: ( a ) Definition of the image-based folding coordinate Z3 . ( b ) Schematic of the single-well to double-well transition in the free energy landscape and the corresponding increments near the origin. ( c ) Experimental signatures of folding reorganization. Symbols show the odd component of the conditional finite-lag mean increment of Z3 . Error bars indicate uncertainty in the estimated response. Curves show polynomial fits; triangles mark zero crossings interpolated from the binned data at 40 V. Insets show apparent potentials obtained as negative antiderivatives of the fitted polynomials.
Figure 4: Correlations of the PCA coordinates. Top: Z2 versus the signed axial endpoint difference, r0,x−rN,x . Bottom: Z3 versus the signed foldedness, (r0,x+rN,x)/2−rcom,x . Correlation coefficients are computed separately for each Wi interval.
Figure 5: Correlations of the image-based coordinates. The image-based coordinate Z~3 is extracted from images generated from simulated polymer conformations, whereas the microscopic folding asymmetry x1+xN−2xcom are computed directly from the simulated configurations.
Figure S1: Transient stretching from the fast, medium, and slow initial conformations (column groups) at different Wi (rows). Each group compares the mean and standard deviation of Z1 and the distribution of nu defined in Eq. ( S23 ) obtained from microscopic simulations (blue) and the conditional UIDD model (orange), using 500 realizations per ensemble. Horizontal axes are given in output-step units.
Figure S2: Experimental response checks and sensitivity to analysis choices. (a,b) Raw conditional response f (gray circles) and odd component g (red squares) at 20 and 40 V. (c) Central slopes s0 at 20 and 40 V for the 72 admissible analysis settings.
Molecular dynamics simulations proceed by integrating the Langevin equations over many small femtosecond timesteps. This poses a challenge for estimating ensemble properties and transition dynamics that occur on much longer timescales. We introduce Langevin Flow Maps, which extend machine-learned force-fields to additionally learn the stochastic Langevin integrator. We show that Langevin Flow Maps enable large-timestep molecular dynamics and recover accurate dynamical properties of the system, while running an order of magnitude faster than current machine-learned force fields. Further, by training on a diverse molecular dataset, we demonstrate a path towards transferable Langevin Flow Maps.
Exact stochastic equations for non-equilibrium dynamics are rarely accessible. We show that the long-time evolution of stochastic dynamics can be predicted from configuration pairs at a fixed short time lag, without knowledge of the equation of motion. Generative diffusion models learn the finite-time transition kernel from these pairs, and iterating it propagates the dynamics far beyond the training lag. For two-dimensional Model B, the diffusive dynamics of a conserved order parameter, the learned kernels reproduce dynamic critical scaling and self-similar t1/3 coarsening. Agreement with direct simulations persists on lattices twice the largest training size and for initial ensembles absent from training. For driven colloids in a periodic optical potential, ten minutes of measured trajectories suffice to predict the particle current and mean passage time over the next twenty minutes within experimental uncertainty. Short-time observations thus contain the information needed to predict emergent non-equilibrium dynamics at much longer times.
Yang-yang Tan, Jinyang Li, Lingxiao Wang
Institute for Physics of Intelligence, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan · RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), Wako, Saitama 351-0198, Japan · KEK Theory Center, Institute of Particle and Nuclear Studies +1
The remarkable ability of modern neural networks to generalize improves with increasing network capacity, even when the number of model parameters or effective degrees of freedom exceeds the number of training data points. This phenomenon is all the more surprising given that generalization error diverges when the number of model parameters approaches a critical value from below. Here we use dynamical mean field theory to show that this so-called "double descent" behavior is the outcome of a phase transition in the stochastic field theory describing the training process. We calculate the critical exponents and scaling function of the double descent phase transition, and show that it is marked by a breakdown of the fluctuation-dissipation theorem associated with broken ergodicity. The corresponding response function has the same functional form as the simple London model of the superconducting transition, with the rigidity of the wave function corresponding to the neural network's ability to generalize accurately.
Chan Li, Nigel Goldenfeld
Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA. · Halıcıoğlu Data Science Institute, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA