Jonah Botvinick-Greenhouse

Jonah Botvinick-Greenhouse, Yunan Yang, Romit Maulik

Motivated by the computational difficulties incurred by popular deep learning algorithms for the generative modeling of temporal densities, we propose a cheap alternative which requires minimal hyperparameter tuning and scales favorably to high dimensional problems. In particular, we use a projection-based optimal transport solver [Meng et al., 2019] to join successive samples and subsequently use transport splines [Chewi et al., 2020] to interpolate the evolving density. When the sampling frequency is sufficiently high, the optimal maps are close to the identity and are thus computationally efficient to compute. Moreover, the training process is highly parallelizable as all optimal maps are independent and can thus be learned simultaneously. Finally, the approach is based solely on numerical linear algebra rather than minimizing a nonconvex objective function, allowing us to easily analyze and control the algorithm. We present several numerical experiments on both synthetic and real-world datasets to demonstrate the efficiency of our method. In particular, these experiments show that the proposed approach is highly competitive compared with state-of-the-art normalizing flows conditioned on time across a wide range of dimensionalities.

Jonah Botvinick-Greenhouse, Maria Oprea, Romit Maulik et al.

The celebrated Takens' embedding theorem provides a theoretical foundation for reconstructing the full state of a dynamical system from partial observations. However, the classical theorem assumes that the underlying system is deterministic and that observations are noise-free, limiting its applicability in real-world scenarios. Motivated by these limitations, we formulate a measure-theoretic generalization that adopts an Eulerian description of the dynamics and recasts the embedding as a pushforward map between spaces of probability measures. Our mathematical results leverage recent advances in optimal transport. Building on the proposed measure-theoretic time-delay embedding theory, we develop a computational procedure that aims to reconstruct the full state of a dynamical system from time-lagged partial observations, engineered with robustness to handle sparse and noisy data. We evaluate our measure-based approach across several numerical examples, ranging from the classic Lorenz-63 system to real-world applications such as NOAA sea surface temperature reconstruction and ERA5 wind field reconstruction.

Jonah Botvinick-Greenhouse, Yunan Yang

We establish guarantees for the unique recovery of vector fields and transport maps from finite measure-valued data, yielding new insights into generative models, data-driven dynamical systems, and PDE inverse problems. In particular, we provide general conditions under which a diffeomorphism can be uniquely identified from its pushforward action on finitely many densities, i.e., when the data $\{(ρ_j,f_\#ρ_j)\}_{j=1}^m$ uniquely determines $f$. As a corollary, we introduce a new metric which compares diffeomorphisms by measuring the discrepancy between finitely many pushforward densities in the space of probability measures. We also prove analogous results in an infinitesimal setting, where derivatives of the densities along a smooth vector field are observed, i.e., when $\{(ρ_j,\text{div} (ρ_j v))\}_{j=1}^m$ uniquely determines $v$. Our analysis makes use of the Whitney and Takens embedding theorems, which provide estimates on the required number of densities $m$, depending only on the intrinsic dimension of the problem. We additionally interpret our results through the lens of Perron--Frobenius and Koopman operators and demonstrate how our techniques lead to new guarantees for the well-posedness of certain PDE inverse problems related to continuity, advection, Fokker--Planck, and advection-diffusion-reaction equations. Finally, we present illustrative numerical experiments demonstrating the unique identification of transport maps from finitely many pushforward densities, and of vector fields from finitely many weighted divergence observations.

Jonah Botvinick-Greenhouse, Wael H. Ali, Mouhacine Benosman et al.

We introduce adaptive-basis physics-informed neural networks (AB-PINNs), a novel approach to domain decomposition for training PINNs in which existing subdomains dynamically adapt to the intrinsic features of the unknown solution. Drawing inspiration from classical mesh refinement techniques, we also modify the domain decomposition on-the-fly throughout training by introducing new subdomains in regions of high residual loss, thereby providing additional expressive power where the solution of the differential equation is challenging to represent. Our flexible approach to domain decomposition is well-suited for multiscale problems, as different subdomains can learn to capture different scales of the underlying solution. Moreover, the ability to introduce new subdomains during training helps prevent convergence to unwanted local minima and can reduce the need for extensive hyperparameter tuning compared to static domain decomposition approaches. Throughout, we present comprehensive numerical results which demonstrate the effectiveness of AB-PINNs at solving a variety of complex multiscale partial differential equations.

Jonah Botvinick-Greenhouse

We propose a novel approach for performing dynamical system identification, based upon the comparison of simulated and observed physical invariant measures. While standard methods adopt a Lagrangian perspective by directly treating time-trajectories as inference data, we take on an Eulerian perspective and instead seek models fitting the observed global time-invariant statistics. With this change in perspective, we gain robustness against pervasive challenges in system identification including noise, chaos, and slow sampling. In the first half of this paper, we pose the system identification task as a partial differential equation (PDE) constrained optimization problem, in which synthetic stationary solutions of the Fokker-Planck equation, obtained as fixed points of a finite-volume discretization, are compared to physical invariant measures extracted from observed trajectory data. In the latter half of the paper, we improve upon this approach in two crucial directions. First, we develop a Galerkin-inspired modification to the finite-volume surrogate model, based on data-adaptive unstructured meshes and Monte-Carlo integration, enabling the approach to efficiently scale to high-dimensional problems. Second, we leverage Takens' seminal time-delay embedding theory to introduce a critical data-dependent coordinate transformation which can guarantee unique system identifiability from the invariant measure alone. This contribution resolves a major challenge of system identification through invariant measures, as systems exhibiting distinct transient behaviors may still share the same time-invariant statistics in their state-coordinates. Throughout, we present comprehensive numerical tests which highlight the effectiveness of our approach on a variety of challenging system identification tasks.

Jonah Botvinick-Greenhouse, Robert Martin, Yunan Yang

While invariant measures are widely employed to analyze physical systems when a direct study of pointwise trajectories is intractable, e.g., due to chaos or noise, they cannot uniquely identify the underlying dynamics. Our first result shows that, in contrast to invariant measures in state coordinates, e.g., $[x(t), y(t), z(t)]$, the invariant measure expressed in time-delay coordinates, e.g., $[x(t), x(t-τ),\ldots, x(t-(m-1)τ)]$, can identify the dynamics up to a topological conjugacy. Our second result resolves the remaining ambiguity: by combining invariant measures constructed from multiple delay frames with distinct observables, the system is uniquely identifiable, provided that a suitable initial condition is satisfied. These guarantees require informative observables and appropriate delay parameters ($m,τ$), which can be limiting in certain settings. We support our theoretical contributions through a series of physical examples demonstrating how invariant measures expressed in delay-coordinates can be used to perform robust system identification in practice.