Danyal Rehman

Grégoire Mialon, Quentin Garrido, Hannah Lawrence et al.

Machine learning for differential equations paves the way for computationally efficient alternatives to numerical solvers, with potentially broad impacts in science and engineering. Though current algorithms typically require simulated training data tailored to a given setting, one may instead wish to learn useful information from heterogeneous sources, or from real dynamical systems observations that are messy or incomplete. In this work, we learn general-purpose representations of PDEs from heterogeneous data by implementing joint embedding methods for self-supervised learning (SSL), a framework for unsupervised representation learning that has had notable success in computer vision. Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs. Code: https://github.com/facebookresearch/SSLForPDEs.

Danyal Rehman, John H. Lienhard

Continuum models for ion transport through polyamide nanopores require solving partial differential equations (PDEs) through complex pore geometries. Resolving spatiotemporal features at this length and time-scale can make solving these equations computationally intractable. In addition, mechanistic models frequently require functional relationships between ion interaction parameters under nano-confinement, which are often too challenging to measure experimentally or know a priori. In this work, we develop the first physics-informed deep learning model to learn ion transport behaviour across polyamide nanopores. The proposed architecture leverages neural differential equations in conjunction with classical closure models as inductive biases directly encoded into the neural framework. The neural differential equations are pre-trained on simulated data from continuum models and fine-tuned on independent experimental data to learn ion rejection behaviour. Gaussian noise augmentations from experimental uncertainty estimates are also introduced into the measured data to improve model generalization. Our approach is compared to other physics-informed deep learning models and shows strong agreement with experimental measurements across all studied datasets.

Danyal Rehman, Tara Akhound-Sadegh, Artem Gazizov et al.

Scalable sampling of molecular states in thermodynamic equilibrium is a long-standing challenge in statistical physics. Boltzmann Generators tackle this problem by pairing a generative model, capable of exact likelihood computation, with importance sampling to obtain consistent samples under the target distribution. Current Boltzmann Generators primarily use continuous normalizing flows (CNFs) trained with flow matching for efficient training of powerful models. However, likelihood calculation for these models is extremely costly, requiring thousands of function evaluations per sample, severely limiting their adoption. In this work, we propose Few-step Accurate Likelihoods for Continuous Flows (FALCON), a method which allows for few-step sampling with a likelihood accurate enough for importance sampling applications by introducing a hybrid training objective that encourages invertibility. We show FALCON outperforms state-of-the-art normalizing flow models for molecular Boltzmann sampling and is two orders of magnitude faster than the equivalently performing CNF model.

Danyal Rehman, John H. Lienhard

Species transport models typically combine partial differential equations (PDEs) with relations from hindered transport theory to quantify electromigrative, convective, and diffusive transport through complex nanoporous systems; however, these formulations are frequently substantial simplifications of the governing dynamics, leading to the poor generalization performance of PDE-based models. Given the growing interest in deep learning methods for the physical sciences, we develop a machine learning-based approach to characterize ion transport across nanoporous membranes. Our proposed framework centers around attention-enhanced neural differential equations that incorporate electroneutrality-based inductive biases to improve generalization performance relative to conventional PDE-based methods. In addition, we study the role of the attention mechanism in illuminating physically-meaningful ion-pairing relationships across diverse mixture compositions. Further, we investigate the importance of pre-training on simulated data from PDE-based models, as well as the performance benefits from hard vs. soft inductive biases. Our results indicate that physics-informed deep learning solutions can outperform their classical PDE-based counterparts and provide promising avenues for modelling complex transport phenomena across diverse applications.

Danyal Rehman, Oscar Davis, Jiarui Lu et al.

Simulation-free training frameworks have been at the forefront of the generative modelling revolution in continuous spaces, leading to large-scale diffusion and flow matching models. However, such modern generative models suffer from expensive inference, inhibiting their use in numerous scientific applications like Boltzmann Generators (BGs) for molecular conformations that require fast likelihood evaluation. In this paper, we revisit classical normalizing flows in the context of BGs that offer efficient sampling and likelihoods, but whose training via maximum likelihood is often unstable and computationally challenging. We propose Regression Training of Normalizing Flows (RegFlow), a novel and scalable regression-based training objective that bypasses the numerical instability and computational challenge of conventional maximum likelihood training in favour of a simple $\ell_2$-regression objective. Specifically, RegFlow maps prior samples under our flow to targets computed using optimal transport couplings or a pre-trained continuous normalizing flow (CNF). To enhance numerical stability, RegFlow employs effective regularization strategies such as a new forward-backward self-consistency loss that enjoys painless implementation. Empirically, we demonstrate that RegFlow unlocks a broader class of architectures that were previously intractable to train for BGs with maximum likelihood. We also show RegFlow exceeds the performance, computational cost, and stability of maximum likelihood training in equilibrium sampling in Cartesian coordinates of alanine dipeptide, tripeptide, and tetrapeptide, showcasing its potential in molecular systems.