Unlike a standard PINN--which produces an approximate Deep neural networks (DNNs) or multi-layer perceptronssolution by minimizing a PDE-residual loss and thus yields (MLPs) offer various inherent advantages over traditionalonly a point estimate, failing to quantify uncertainty inapproaches of scientific computing and data analysis, suchduced by noisy or limited data, a Bayesian PINN returns a as finite element methods, wavelets and kernel methods, full posterior distribution over solutions by combining the which are often hampered by the irregular and nonlinearuncertain information from the likelihood (data) and the data structures and the high input dimensions. In contrast, DNNs are capable of approximating a rich class of functions prior. Bayesian neural networks, originating in the seminal works of MacKay (MacKay, 1995) and Neal (Neal, 1995), with aforementioned complexities and can also easily en-have been extensively studied over the past three decades codes additional complex physical structures, such as sym- (Lampinen & Vehtari, 2001; Titterington, 2004; Graves, metry and other invariant structures.

Label efficiency has become an increasingly important objective in deep learning applications. Active learning aims to reduce the number of labeled examples needed to train deep networks, but the empirical performance of active learning algorithms can vary dramatically across datasets and applications. It is difficult to know in advance which active learning strategy will perform well or best in a given application. To address this, we propose the first adaptive algorithm selection strategy for deep active learning. For any unlabeled dataset, our (meta) algorithm TAILOR(Thompson ActIve Learning algORithm selection) iteratively and adaptively chooses among a set of candidate active learning algorithms. TAILORuses novel reward functions aimed at gathering class-balanced examples. Extensive experiments in multi-class and multi-label applications demonstrate TAILOR's effectiveness in achieving accuracy comparable or better than that of the best of the candidate algorithms. Our implementation of TAILOR is open-sourced at https://github.com/jifanz/TAILOR.

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.

Continual learning (CL) empowers pre-trained vision-language (VL) models to efficiently adapt to a sequence of downstream tasks.

The space ofdifferential k-forms (also calledk-forms in short), denotedAk(Rn), is defined by smoothlyassigningtoeachx Rn ak-linearalternatingformw Λk(Rn).

ImageNetC measures the robustness against common corruptions such as Gaussian noise, blur, or weather changes. We follow Hendrycks and Dietterich[10] for measuring mean corruption error (mCE). The lower ImageNet-C implies that themodel isrobustagainst corruption noises. BGC evaluates the robustness against background manipulations aswell asthe adversarial robustness.

Weintroduce Unbalanced SobolevDescent (USD), aparticle descent algorithm for transporting a high dimensional source distribution to a target distribution that does not necessarily have the same mass.

Gradient descent: Asillustrated byR1'sexample off(x),ourcorrectness condition forautodiffsystems doesnot12 necessarily imply the correctness of the gradient descent based on those systems (i.e., that the gradient descent13 converges to Clarke critical points). This gives a partial answer to R3's question on possible drawbacks of using14 intensionalderivatives. This is a good question that would lead to interesting future work.

In the following, we provide a brief overview of Riemannian geometry and constant curvature manifolds, specifically the Poincaré ball and the hypersphere models. Sphere In the two-dimensional settingd = 2, we rely on polar coordinates to parametrize the sphere S2. In the following subsection we remind that this regularization term can also be motivated from an estimator'svarianceperspective. 5 D.2 Frobeniusnorm Hutchinson'sestimator Hutchinson'sestimator(Hutchinson,1990)isasimple waytoobtain a stochastic estimate ofthetrace ofamatrix. The variance of this estimator thus depends on the Frobenius norm of the vector's field Jacobian Thenγ(tn) is also a Cauchy sequence by Equation 16. So for every sequence (tn) in (a,b) that converges tob, we have that(γ(tn)) converges top.