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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: //github.com/facebookresearch/SSLForPDEs.

We propose Structured Language Generation Model (SLGM), a mixture of new loss function and inference method for better generalization of structured outputs. Previous studies on structure prediction (e.g. NER, RE) make use of explicit dataset information, which would boost performance, yet it might pose challenges to robust generalization in real-world situations. Instead, our model gives generalized format information about data indirectly. With format information, we could reduce sequence-to-sequence problem into classification problem via loss calibration and formatted decoding. Our experimental results showed SLGM successfully maintain performance without dataset information, and showed much less format errors. We also showed our model can work like adapters on individual dataset, with no additional training.

We conduct a systematic study of the approximation properties of Transformer for sequence modeling with long, sparse and complicated memory. We investigate the mechanisms through which different components of Transformer, such as the dot-product self-attention, positional encoding and feed-forward layer, affect its expressive power, and we study their combined effects through establishing explicit approximation rates. Our study reveals the roles of critical parameters in the Transformer, such as the number of layers and the number of attention heads, and these insights also provide natural suggestions for alternative architectures.

Given an edge-weighted metric complete graph with $n$ vertices, the maximum weight metric triangle packing problem is to find a set of $n/3$ vertex-disjoint triangles with the total weight of all triangles in the packing maximized. Several simple methods can lead to a 2/3-approximation ratio. However, this barrier is not easy to break. Chen et al. proposed a randomized approximation algorithm with an expected ratio of $(0.66768-\varepsilon)$ for any constant $\varepsilon>0$. In this paper, we improve the approximation ratio to $(0.66835-\varepsilon)$. Furthermore, we can derandomize our algorithm.

In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a `base' of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems K, KT , K4, and S4, with $\square$ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between $\square$ and a natural presentation of $\lozenge$. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases.

Bayesian optimization has been successfully applied to optimize black-box functions where the number of evaluations is severely limited. However, in many real-world applications, it is hard or impossible to know in advance which designs are feasible due to some physical or system limitations. These issues lead to an even more challenging problem of optimizing an unknown function with unknown constraints. In this paper, we observe that in such scenarios optimal solution typically lies on the boundary between feasible and infeasible regions of the design space, making it considerably more difficult than that with interior optima. Inspired by this observation, we propose BE-CBO, a new Bayesian optimization method that efficiently explores the boundary between feasible and infeasible designs. To identify the boundary, we learn the constraints with an ensemble of neural networks that outperform the standard Gaussian Processes for capturing complex boundaries. Our method demonstrates superior performance against state-of-the-art methods through comprehensive experiments on synthetic and real-world benchmarks.

Ordinary differential equations (ODEs) underlie dynamical systems which serve as models for a vast number of natural and social phenomena. Yet inferring the ODE that best describes a set of noisy observations on one such phenomenon can be remarkably challenging, and the models available to achieve it tend to be highly specialized and complex too. In this work we propose a novel supervised learning framework for zero-shot inference of ODEs from noisy data. We first generate large datasets of one-dimensional ODEs, by sampling distributions over the space of initial conditions, and the space of vector fields defining them. We then learn neural maps between noisy observations on the solutions of these equations, and their corresponding initial condition and vector fields. The resulting models, which we call foundational inference models (FIM), can be (i) copied and matched along the time dimension to increase their resolution; and (ii) copied and composed to build inference models of any dimensionality, without the need of any finetuning. We use FIM to model both ground-truth dynamical systems of different dimensionalities and empirical time series data in a zero-shot fashion, and outperform state-of-the-art models which are finetuned to these systems. Our (pretrained) FIMs are available online

Score-based generative models are a popular class of generative modelling techniques relying on stochastic differential equations (SDE). From their inception, it was realized that it was also possible to perform generation using ordinary differential equations (ODE) rather than SDE. This led to the introduction of the probability flow ODE approach and denoising diffusion implicit models. Flow matching methods have recently further extended these ODE-based approaches and approximate a flow between two arbitrary probability distributions. Previous work derived bounds on the approximation error of diffusion models under the stochastic sampling regime, given assumptions on the $L^2$ loss. We present error bounds for the flow matching procedure using fully deterministic sampling, assuming an $L^2$ bound on the approximation error and a certain regularity condition on the data distributions.

We consider the task of constructing confidence intervals with differential privacy. We propose two private variants of the non-parametric bootstrap, which privately compute the median of the results of multiple ``little'' bootstraps run on partitions of the data and give asymptotic bounds on the coverage error of the resulting confidence intervals. For a fixed differential privacy parameter $\epsilon$, our methods enjoy the same error rates as that of the non-private bootstrap to within logarithmic factors in the sample size $n$. We empirically validate the performance of our methods for mean estimation, median estimation, and logistic regression with both real and synthetic data. Our methods achieve similar coverage accuracy to existing methods (and non-private baselines) while providing notably shorter ($\gtrsim 10$ times) confidence intervals than previous approaches.

We develop a general theory to optimize the frequentist regret for sequential learning problems, where efficient bandit and reinforcement learning algorithms can be derived from unified Bayesian principles. We propose a novel optimization approach to generate "algorithmic beliefs" at each round, and use Bayesian posteriors to make decisions. The optimization objective to create "algorithmic beliefs," which we term "Algorithmic Information Ratio," represents an intrinsic complexity measure that effectively characterizes the frequentist regret of any algorithm. To the best of our knowledge, this is the first systematical approach to make Bayesian-type algorithms prior-free and applicable to adversarial settings, in a generic and optimal manner. Moreover, the algorithms are simple and often efficient to implement. As a major application, we present a novel algorithm for multi-armed bandits that achieves the "best-of-all-worlds" empirical performance in the stochastic, adversarial, and non-stationary environments. And we illustrate how these principles can be used in linear bandits, bandit convex optimization, and reinforcement learning.