Hao Wu

Recovering unbiased kinetic and thermodynamic observables from enhanced sampling simulations is a central challenge in rare-event sampling. Classical Girsanov Reweighting (GR) offers a principled solution by yielding exact pathwise probability ratios between biased and unbiased processes. However, the variance of GR weights grows rapidly with time, rendering it impractical for long-horizon reweighting. We introduce Marginal Girsanov Reweighting (MGR), which mitigates variance explosion by marginalizing over intermediate paths, producing stable and scalable weights for long-time-scale dynamics. Experiments on various molecular dynamics (MD) systems demonstrate that MGR accurately recovers unbiased kinetic properties from trajectories generated under biased simulations.

We introduce π-Girsanov, a new method for constructing Markov state models from biased enhanced-sampling molecular dynamics simulations based on Girsanov reweighting. The key idea behind this new method is to separate the reweighting of the stationary density from the reweighting of the correlation function. We evaluate the effectiveness of this approach on several analytical potentials and on a model biomolecular system, comparing its performance with the original method. Our results show that π-Girsanov not only improves the estimation in a single-ensemble setting but also resolves key challenges in estimating transition matrices from multiensemble and non-equilibrium biased trajectories. Overall, π-Girsanov represents a substantial advance in kinetic reweighting, strengthening the connection between enhanced sampling techniques and Markov state modeling.

Physics-informed operator learning provides an efficient framework for approximating solution operators of partial differential equations by combining observational data with governing physical laws. However, most existing methods implicitly assume that the prescribed governing equation is accurate. This assumption may fail in practical applications, where model simplifications, missing physical effects, parameter drift, or incomplete constitutive relations can lead to model misspecification. In this work, we propose a physics-guided operator correction framework for learning solution operators under misspecified governing equations. At the operator level, the target mapping is decomposed into a prior operator induced by an approximate physical model and a learnable correction operator that accounts for the remaining discrepancy. We realize the operator model correction using a serial DeepONet architecture, where the first DeepONet provides a solution prediction and the second DeepONet learns an additive correction conditioned on both the input function and the solution prediction. The learned correction is incorporated into the physics residual and trained together with data-consistency constraints, allowing the model to retain useful physical structure while adapting to inaccurate governing equations. Numerical experiments on diffusion-reaction, Burgers, cavity flow, and hyperelastic problems show that the proposed method substantially reduces errors induced by misspecified physics. Additional tests under sparse and noisy observations further demonstrate the robustness of the framework and its ability to provide informative uncertainty estimates through deep ensembles.

Model correction is essential for reliable PDE learning under misspecified physics arising from simplified assumptions or limited observations. Most existing correction methods operate in parameter space, where uncertainty quantification typically requires sampling or ensembles and can be prohibitively expensive. To address this challenge, we propose a representation-level, latent-space correction framework by extending LVM-GP, a probabilistic PDE solver that couples a latent variable model with Gaussian processes. Our method uses a shared confidence-aware encoder with two probabilistic decoders to jointly model the solution and a discrepancy correction. The encoder learns a stochastic latent representation by interpolating between deterministic features and a GP prior via a learnable confidence function. Conditioned on this shared latent representation, the two decoders jointly quantify uncertainty in both the solution and the correction under soft physics constraints with noisy data. A latent-space regularizer further structures the representation and improves robustness. This unified design enables joint uncertainty quantification of the solution and the correction within a single training procedure, without parameter sampling or repeated retraining. Numerical experiments show accuracy comparable to Ensemble PINNs and B-PINNs, with improved computational efficiency and robustness to misspecified physics.

Model correction is essential for reliable PDE learning under misspecified physics arising from simplified assumptions or limited observations. Most existing correction methods operate in parameter space, where uncertainty quantification typically requires sampling or ensembles and can be prohibitively expensive. To address this challenge, we propose a representation-level, latent-space correction framework by extending LVM-GP, a probabilistic PDE solver that couples a latent variable model with Gaussian processes. Our method uses a shared confidence-aware encoder with two probabilistic decoders to jointly model the solution and a discrepancy correction. The encoder learns a stochastic latent representation by interpolating between deterministic features and a GP prior via a learnable confidence function. Conditioned on this shared latent representation, the two decoders jointly quantify uncertainty in both the solution and the correction under soft physics constraints with noisy data. A latent-space regularizer further structures the representation and improves robustness. This unified design enables joint uncertainty quantification of the solution and the correction within a single training procedure, without parameter sampling or repeated retraining. Numerical experiments show accuracy comparable to Ensemble PINNs and B-PINNs, with improved computational efficiency and robustness to misspecified physics.

We propose a novel probabilistic framework, termed LVM-GP, for uncertainty quantification in solving forward and inverse partial differential equations (PDEs) with noisy data. The core idea is to construct a stochastic mapping from the input to a high-dimensional latent representation, enabling uncertainty-aware prediction of the solution. Specifically, the architecture consists of a confidence-aware encoder and a probabilistic decoder. The encoder implements a high-dimensional latent variable model based on a Gaussian process (LVM-GP), where the latent representation is constructed by interpolating between a learnable deterministic feature and a Gaussian process prior, with the interpolation strength adaptively controlled by a confidence function learned from data. The decoder defines a conditional Gaussian distribution over the solution field, where the mean is predicted by a neural operator applied to the latent representation, allowing the model to learn flexible function-to-function mapping. Moreover, physical laws are enforced as soft constraints in the loss function to ensure consistency with the underlying PDE structure. Compared to existing approaches such as Bayesian physics-informed neural networks (B-PINNs) and deep ensembles, the proposed framework can efficiently capture functional dependencies via merging a latent Gaussian process and neural operator, resulting in competitive predictive accuracy and robust uncertainty quantification. Numerical experiments demonstrate the effectiveness and reliability of the method.

Locating the transition states (TS) for the conformational changes of biomacromolecules is among the major tasks of biomolecular simulations, as they are the bottlenecks of motion encoding key mechanistic insights. However, identifying the short-lived TSs from (even abundant) simulation data has been a long-standing challenge due to the high dimensionality of the molecules. Gentlest ascent dynamics (GAD) is an effective approach that searches for saddle points but only within spaces of small number of (typically <20) dimensions. Such a restriction of GAD may in principle be relieved by dimensionality reduction (DR) that reduces the high-dimensional configurational space of the molecules to a low-dimensional manifold. However, the vast majority of DR algorithms are built to focus on only high-density regions and have therefore distorted the TS regions, disabling a subsequent GAD search. The recently introduced reaction coordinate flows (RCF) is among the few exceptions. As RCF learns an invertible mapping between the configurational space and the reduced RC space through a loss function incorporating both density and transition pair information, it shall be able to preserve kinetics and therefore TS during DR. GAD can then be readily applied to locate the TS candidates in the RCF-learned RC space, which can be validated rigorously through reverse RCF mapping and committor analysis in the original space. Here, we demonstrate the effectiveness of this RCF-GAD integration through alanine dipeptide and the ground-to-excited transition of the T4 lysozyme L99A variant (T4L-L99A) in explicit solvents. For alanine dipeptide, GAD managed to identify three TSs with the RCF reduction to four RCs, but only two TSs with a reduction to two RCs, due to the merging of two low density stable states in the 2RC representation, indicate the necessity of a priori evaluation of the number of intrinsic dimensions for RCF. For T4L-L99A, the TSs located by GAD in a 4RC RCF reduction successfully resembled those found previously via automated path searching, demonstrating the feasibility of our approach for realistic biomolecular systems.