VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Prediction

In-Context Operator Networks (ICONs) are models that learn operators across different types of PDEs using a few-shot, in-context approach. Although they show successful generalization to various PDEs, existing methods treat each data point as a single token, and suffer from computational inefficiency when processing dense data, limiting their application in higher spatial dimensions. In this work, we propose Vision In-Context Operator Networks (VICON), incorporating a vision transformer architecture that efficiently processes 2D functions through patch-wise operations. We evaluated our method on three fluid dynamics datasets, demonstrating both superior performance (reducing scaled $L^2$ error by $40\%$ and $61.6\%$ for two benchmark datasets for compressible flows, respectively) and computational efficiency (requiring only one-third of the inference time per frame) in long-term rollout predictions compared to the current state-of-the-art sequence-to-sequence model with fixed timestep prediction: Multiple Physics Pretraining (MPP). Compared to MPP, our method preserves the benefits of in-context operator learning, enabling flexible context formation when dealing with insufficient frame counts or varying timestep values.

A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equations

We propose a scalable preconditioned primal-dual hybrid gradient algorithm for solving partial differential equations (PDEs). We multiply the PDE with a dual test function to obtain an inf-sup problem whose loss functional involves lower-order differential operators. The Primal-Dual Hybrid Gradient (PDHG) algorithm is then leveraged for this saddle point problem. By introducing suitable precondition operators to the proximal steps in the PDHG algorithm, we obtain an alternative natural gradient ascent-descent optimization scheme for updating the neural network parameters. We apply the Krylov subspace method (MINRES) to evaluate the natural gradients efficiently. Such treatment readily handles the inversion of precondition matrices via matrix-vector multiplication. A posterior convergence analysis is established for the time-continuous version of the proposed method. The algorithm is tested on various types of PDEs with dimensions ranging from $1$ to $50$, including linear and nonlinear elliptic equations, reaction-diffusion equations, and Monge-Amp\`ere equations stemming from the $L^2$ optimal transport problems. We compare the performance of the proposed method with several commonly used deep learning algorithms such as physics-informed neural networks (PINNs), the DeepRitz method, weak adversarial networks (WANs), etc, for solving PDEs using the Adam and L-BFGS optimizers. The numerical results suggest that the proposed method performs efficiently and robustly and converges more stably.

Fried deconvolution

In this paper we present a new approach to deblur the effect of atmospheric turbulence in the case of long range imaging. Our method is based on an analytical formulation, the Fried kernel, of the atmosphere modulation transfer function (MTF) and a framelet based deconvolution algorithm. An important parameter is the refractive index structure which requires specific measurements to be known. Then we propose a method which provides a good estimation of this parameter from the input blurred image. The final algorithms are very easy to implement and show very good results on both simulated blur and real images.

2D Empirical Transforms. Wavelets, Ridgelets and Curvelets revisited

A recently developed new approach, called ``Empirical Wavelet Transform'', aims to build 1D adaptive wavelet frames accordingly to the analyzed signal. In this paper, we present several extensions of this approach to 2D signals (images). We revisit some well-known transforms (tensor wavelets, Littlewood-Paley wavelets, ridgelets and curvelets) and show that it is possible to build their empirical counterpart. We prove that such constructions lead to different adaptive frames which show some promising properties for image analysis and processing.

Bregman implementation of Meyer's $G-$norm for cartoon + textures decomposition

In this paper, we design a very simple algorithm based on Split Bregman iterations to numerically solve the cartoon + textures decomposition model of Meyer. This results in a significant gain in speed compared to Chambolle's nonlinear projectors.

Wavelet Burst Accumulation for turbulence mitigation

In this paper, we investigate the extension of the recently proposed weighted Fourier burst accumulation (FBA) method into the wavelet domain. The purpose of FBA is to reconstruct a clean and sharp image from a sequence of blurred frames. This concept lies in the construction of weights to amplify dominant frequencies in the Fourier spectrum of each frame. The reconstructed image is then obtained by taking the inverse Fourier transform of the average of all processed spectra. In this paper, we first suggest to replace the rigid registration step used in the original algorithm by a non-rigid registration in order to be able to process sequences acquired through atmospheric turbulence. Second, we propose to work in a wavelet domain instead of the Fourier one. This leads us to the construction of two types of algorithms. Finally, we propose an alternative approach to replace the weighting idea by an approach promoting the sparsity in the used space. Several experiments are provided to illustrate the efficiency of the proposed methods.

Score-based Neural Ordinary Differential Equations for Computing Mean Field Control Problems

Classical neural ordinary differential equations (ODEs) are powerful tools for approximating the log-density functions in high-dimensional spaces along trajectories, where neural networks parameterize the velocity fields. This paper proposes a system of neural differential equations representing first- and second-order score functions along trajectories based on deep neural networks. We reformulate the mean field control (MFC) problem with individual noises into an unconstrained optimization problem framed by the proposed neural ODE system. Additionally, we introduce a novel regularization term to enforce characteristics of viscous Hamilton--Jacobi--Bellman (HJB) equations to be satisfied based on the evolution of the second-order score function. Examples include regularized Wasserstein proximal operators (RWPOs), probability flow matching of Fokker--Planck (FP) equations, and linear quadratic (LQ) MFC problems, which demonstrate the effectiveness and accuracy of the proposed method.

Noise-Free Sampling Algorithms via Regularized Wasserstein Proximals

We consider the problem of sampling from a distribution governed by a potential function. This work proposes an explicit score-based MCMC method that is deterministic, resulting in a deterministic evolution for particles rather than a stochastic differential equation evolution. The score term is given in closed form by a regularized Wasserstein proximal, using a kernel convolution that is approximated by sampling. We demonstrate fast convergence on various problems and show improved dimensional dependence of mixing time bounds for the case of Gaussian distributions compared to the unadjusted Langevin algorithm (ULA) and the Metropolis-adjusted Langevin algorithm (MALA). We additionally derive closed form expressions for the distributions at each iterate for quadratic potential functions, characterizing the variance reduction. Empirical results demonstrate that the particles behave in an organized manner, lying on level set contours of the potential. Moreover, the posterior mean estimator of the proposed method is shown to be closer to the maximum a-posteriori estimator compared to ULA and MALA, in the context of Bayesian logistic regression.

Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO Scheme

A normalizing flow (NF) is a mapping that transforms a chosen probability distribution to a normal distribution. Such flows are a common technique used for data generation and density estimation in machine learning and data science. The density estimate obtained with a NF requires a change of variables formula that involves the computation of the Jacobian determinant of the NF transformation. In order to tractably compute this determinant, continuous normalizing flows (CNF) estimate the mapping and its Jacobian determinant using a neural ODE. Optimal transport (OT) theory has been successfully used to assist in finding CNFs by formulating them as OT problems with a soft penalty for enforcing the standard normal distribution as a target measure. A drawback of OT-based CNFs is the addition of a hyperparameter, $\alpha$, that controls the strength of the soft penalty and requires significant tuning. We present JKO-Flow, an algorithm to solve OT-based CNF without the need of tuning $\alpha$. This is achieved by integrating the OT CNF framework into a Wasserstein gradient flow framework, also known as the JKO scheme. Instead of tuning $\alpha$, we repeatedly solve the optimization problem for a fixed $\alpha$ effectively performing a JKO update with a time-step $\alpha$. Hence we obtain a "divide and conquer" algorithm by repeatedly solving simpler problems instead of solving a potentially harder problem with large $\alpha$.

Improving Generative Flow Networks with Path Regularization

Generative Flow Networks (GFlowNets) are recently proposed models for learning stochastic policies that generate compositional objects by sequences of actions with the probability proportional to a given reward function. The central problem of GFlowNets is to improve their exploration and generalization. In this work, we propose a novel path regularization method based on optimal transport theory that places prior constraints on the underlying structure of the GFlowNets. The prior is designed to help the GFlowNets better discover the latent structure of the target distribution or enhance its ability to explore the environment in the context of active learning. The path regularization controls the flow in GFlowNets to generate more diverse and novel candidates via maximizing the optimal transport distances between two forward policies or to improve the generalization via minimizing the optimal transport distances. In addition, we derive an efficient implementation of the regularization by finding its closed form solutions in specific cases and a meaningful upper bound that can be used as an approximation to minimize the regularization term. We empirically demonstrate the advantage of our path regularization on a wide range of tasks, including synthetic hypergrid environment modeling, discrete probabilistic modeling, and biological sequence design.