Gilles Louppe

Florent Purnode, Louis Denis, François Henrotte et al.

Incorporating hysteresis and eddy currents into finite element simulations of laminated-core electrical machines is computationally challenging. Resolving the fields inside the laminations at each integration point and at every nonlinear iteration leads to computational costs several orders of magnitude higher than anhysteretic simulations, making such approaches impractical for design applications. Conversely, simplified models accounting only for magnetic saturation are becoming increasingly inadequate as electrical machine topologies and operating conditions grow in complexity. In this context, machine learning surrogate modeling has emerged as a promising alternative, offering efficient and accurate approximations of complex electromagnetic behaviors. In this paper, a recurrent neural network is trained as a surrogate of a laminated-core material model for an isotropic laminated core, and is integrated into realistic two-dimensional magnetodynamic finite element simulations based on a magnetic vector potential formulation. The proposed approach achieves excellent agreement with the reference laminated-core model while limiting the computational cost to about twice that of an anhysteretic simulation. By training the recurrent neural network on a sufficiently diverse set of artificially generated magnetic field sequences designed to mimic those encountered in electrical machine simulations, the proposed approach can be readily applied across a wide range of finite element simulations. Furthermore, the trained surrogate model is provided as a standalone component that can be easily incorporated into existing computational frameworks. It is publicly available at https://gitlab.onelab.info/getdp/lamnet.

Renaud Vandeghen, Gilles Louppe, Marc Van Droogenbroeck

Deep learning has emerged as an effective solution for solving the task of object detection in images but at the cost of requiring large labeled datasets. To mitigate this cost, semi-supervised object detection methods, which consist in leveraging abundant unlabeled data, have been proposed and have already shown impressive results. However, most of these methods require linking a pseudo-label to a ground-truth object by thresholding. In previous works, this threshold value is usually determined empirically, which is time consuming, and only done for a single data distribution. When the domain, and thus the data distribution, changes, a new and costly parameter search is necessary. In this work, we introduce our method Adaptive Self-Training for Object Detection (ASTOD), which is a simple yet effective teacher-student method. ASTOD determines without cost a threshold value based directly on the ground value of the score histogram. To improve the quality of the teacher predictions, we also propose a novel pseudo-labeling procedure. We use different views of the unlabeled images during the pseudo-labeling step to reduce the number of missed predictions and thus obtain better candidate labels. Our teacher and our student are trained separately, and our method can be used in an iterative fashion by replacing the teacher by the student. On the MS-COCO dataset, our method consistently performs favorably against state-of-the-art methods that do not require a threshold parameter, and shows competitive results with methods that require a parameter sweep search. Additional experiments with respect to a supervised baseline on the DIOR dataset containing satellite images lead to similar conclusions, and prove that it is possible to adapt the score threshold automatically in self-training, regardless of the data distribution. The code is available at https:// github.com/rvandeghen/ASTOD

François Rozet, Gilles Louppe

Data assimilation, in its most comprehensive form, addresses the Bayesian inverse problem of identifying plausible state trajectories that explain noisy or incomplete observations of stochastic dynamical systems. Various approaches have been proposed to solve this problem, including particle-based and variational methods. However, most algorithms depend on the transition dynamics for inference, which becomes intractable for long time horizons or for high-dimensional systems with complex dynamics, such as oceans or atmospheres. In this work, we introduce score-based data assimilation for trajectory inference. We learn a score-based generative model of state trajectories based on the key insight that the score of an arbitrarily long trajectory can be decomposed into a series of scores over short segments. After training, inference is carried out using the score model, in a non-autoregressive manner by generating all states simultaneously. Quite distinctively, we decouple the observation model from the training procedure and use it only at inference to guide the generative process, which enables a wide range of zero-shot observation scenarios. We present theoretical and empirical evidence supporting the effectiveness of our method.

Arnaud Delaunoy, Joeri Hermans, François Rozet et al.

Modern approaches for simulation-based inference rely upon deep learning surrogates to enable approximate inference with computer simulators. In practice, the estimated posteriors' computational faithfulness is, however, rarely guaranteed. For example, Hermans et al. (2021) show that current simulation-based inference algorithms can produce posteriors that are overconfident, hence risking false inferences. In this work, we introduce Balanced Neural Ratio Estimation (BNRE), a variation of the NRE algorithm designed to produce posterior approximations that tend to be more conservative, hence improving their reliability, while sharing the same Bayes optimal solution. We achieve this by enforcing a balancing condition that increases the quantified uncertainty in small simulation budget regimes while still converging to the exact posterior as the budget increases. We provide theoretical arguments showing that BNRE tends to produce posterior surrogates that are more conservative than NRE's. We evaluate BNRE on a wide variety of tasks and show that it produces conservative posterior surrogates on all tested benchmarks and simulation budgets. Finally, we emphasize that BNRE is straightforward to implement over NRE and does not introduce any computational overhead.

Arnaud Delaunoy, Benjamin Kurt Miller, Patrick Forré et al.

Conservative inference is a major concern in simulation-based inference. It has been shown that commonly used algorithms can produce overconfident posterior approximations. Balancing has empirically proven to be an effective way to mitigate this issue. However, its application remains limited to neural ratio estimation. In this work, we extend balancing to any algorithm that provides a posterior density. In particular, we introduce a balanced version of both neural posterior estimation and contrastive neural ratio estimation. We show empirically that the balanced versions tend to produce conservative posterior approximations on a wide variety of benchmarks. In addition, we provide an alternative interpretation of the balancing condition in terms of the $χ^2$ divergence.

François Rozet, Gilles Louppe

Data assimilation addresses the problem of identifying plausible state trajectories of dynamical systems given noisy or incomplete observations. In geosciences, it presents challenges due to the high-dimensionality of geophysical dynamical systems, often exceeding millions of dimensions. This work assesses the scalability of score-based data assimilation (SDA), a novel data assimilation method, in the context of such systems. We propose modifications to the score network architecture aimed at significantly reducing memory consumption and execution time. We demonstrate promising results for a two-layer quasi-geostrophic model.

Namid R. Stillman, Silke Henkes, Roberto Mayor et al.

Many collective systems exist in nature far from equilibrium, ranging from cellular sheets up to flocks of birds. These systems reflect a form of active matter, whereby individual material components have internal energy. Under specific parameter regimes, these active systems undergo phase transitions whereby small fluctuations of single components can lead to global changes to the rheology of the system. Simulations and methods from statistical physics are typically used to understand and predict these phase transitions for real-world observations. In this work, we demonstrate that simulation-based inference can be used to robustly infer active matter parameters from system observations. Moreover, we demonstrate that a small number (from one to three) snapshots of the system can be used for parameter inference and that this graph-informed approach outperforms typical metrics such as the average velocity or mean square displacement of the system. Our work highlights that high-level system information is contained within the relational structure of a collective system and that this can be exploited to better couple models to data.

Maciej Falkiewicz, Naoya Takeishi, Imahn Shekhzadeh et al.

Bayesian inference allows expressing the uncertainty of posterior belief under a probabilistic model given prior information and the likelihood of the evidence. Predominantly, the likelihood function is only implicitly established by a simulator posing the need for simulation-based inference (SBI). However, the existing algorithms can yield overconfident posteriors (Hermans *et al.*, 2022) defeating the whole purpose of credibility if the uncertainty quantification is inaccurate. We propose to include a calibration term directly into the training objective of the neural model in selected amortized SBI techniques. By introducing a relaxation of the classical formulation of calibration error we enable end-to-end backpropagation. The proposed method is not tied to any particular neural model and brings moderate computational overhead compared to the profits it introduces. It is directly applicable to existing computational pipelines allowing reliable black-box posterior inference. We empirically show on six benchmark problems that the proposed method achieves competitive or better results in terms of coverage and expected posterior density than the previously existing approaches.

Sacha Lewin, Maxime Vandegar, Thomas Hoyoux et al.

The long-standing problem of novel view synthesis has many applications, notably in sports broadcasting. Photorealistic novel view synthesis of soccer actions, in particular, is of enormous interest to the broadcast industry. Yet only a few industrial solutions have been proposed, and even fewer that achieve near-broadcast quality of the synthetic replays. Except for their setup of multiple static cameras around the playfield, the best proprietary systems disclose close to no information about their inner workings. Leveraging multiple static cameras for such a task indeed presents a challenge rarely tackled in the literature, for a lack of public datasets: the reconstruction of a large-scale, mostly static environment, with small, fast-moving elements. Recently, the emergence of neural radiance fields has induced stunning progress in many novel view synthesis applications, leveraging deep learning principles to produce photorealistic results in the most challenging settings. In this work, we investigate the feasibility of basing a solution to the task on dynamic NeRFs, i.e., neural models purposed to reconstruct general dynamic content. We compose synthetic soccer environments and conduct multiple experiments using them, identifying key components that help reconstruct soccer scenes with dynamic NeRFs. We show that, although this approach cannot fully meet the quality requirements for the target application, it suggests promising avenues toward a cost-efficient, automatic solution. We also make our work dataset and code publicly available, with the goal to encourage further efforts from the research community on the task of novel view synthesis for dynamic soccer scenes. For code, data, and video results, please see https://soccernerfs.isach.be.

Aaron David Schneider, Paul Mollière, Gilles Louppe et al.

To understand high precision observations of exoplanets and brown dwarfs, we need detailed and complex general circulation models (GCMs) that incorporate hydrodynamics, chemistry, and radiation. For this study, we specifically examined the coupling between chemistry and radiation in GCMs and compared different methods for the mixing of opacities of different chemical species in the correlated-k assumption, when equilibrium chemistry cannot be assumed. We propose a fast machine learning method based on DeepSets (DS), which effectively combines individual correlated-k opacities (k-tables). We evaluated the DS method alongside other published methods such as adaptive equivalent extinction (AEE) and random overlap with rebinning and resorting (RORR). We integrated these mixing methods into our GCM (expeRT/MITgcm) and assessed their accuracy and performance for the example of the hot Jupiter HD~209458 b. Our findings indicate that the DS method is both accurate and efficient for GCM usage, whereas RORR is too slow. Additionally, we observed that the accuracy of AEE depends on its specific implementation and may introduce numerical issues in achieving radiative transfer solution convergence. We then applied the DS mixing method in a simplified chemical disequilibrium situation, where we modeled the rainout of TiO and VO, and confirmed that the rainout of TiO and VO would hinder the formation of a stratosphere. To further expedite the development of consistent disequilibrium chemistry calculations in GCMs, we provide documentation and code for coupling the DS mixing method with correlated-k radiative transfer solvers. The DS method has been extensively tested to be accurate enough for GCMs; however, other methods might be needed for accelerating atmospheric retrievals.

Norman Marlier, Olivier Brüls, Gilles Louppe

General robotic grippers are challenging to control because of their rich nonsmooth contact dynamics and the many sources of uncertainties due to the environment or sensor noise. In this work, we demonstrate how to compute 6-DoF grasp poses using simulation-based Bayesian inference through the full stochastic forward simulation of the robot in its environment while robustly accounting for many of the uncertainties in the system. A Riemannian manifold optimization procedure preserving the nonlinearity of the rotation space is used to compute the maximum a posteriori grasp pose. Simulation and physical benchmarks show the promising high success rate of the approach.

Victor Mangeleer, Gilles Louppe

In climate simulations, small-scale processes shape ocean dynamics but remain computationally expensive to resolve directly. For this reason, their contributions are commonly approximated using empirical parameterizations, which lead to significant errors in long-term projections. In this work, we develop parameterizations based on Fourier Neural Operators, showcasing their accuracy and generalizability in comparison to other approaches. Finally, we discuss the potential and limitations of neural networks operating in the frequency domain, paving the way for future investigation.

Arnaud Delaunoy, Maxence de la Brassinne Bonardeaux, Siddharth Mishra-Sharma et al.

Simulation-based inference methods have been shown to be inaccurate in the data-poor regime, when training simulations are limited or expensive. Under these circumstances, the inference network is particularly prone to overfitting, and using it without accounting for the computational uncertainty arising from the lack of identifiability of the network weights can lead to unreliable results. To address this issue, we propose using Bayesian neural networks in low-budget simulation-based inference, thereby explicitly accounting for the computational uncertainty of the posterior approximation. We design a family of Bayesian neural network priors that are tailored for inference and show that they lead to well-calibrated posteriors on tested benchmarks, even when as few as $O(10)$ simulations are available. This opens up the possibility of performing reliable simulation-based inference using very expensive simulators, as we demonstrate on a problem from the field of cosmology where single simulations are computationally expensive. We show that Bayesian neural networks produce informative and well-calibrated posterior estimates with only a few hundred simulations.

Norman Marlier, Julien Gustin, Olivier Brüls et al.

Robotic grasping in highly noisy environments presents complex challenges, especially with limited prior knowledge about the scene. In particular, identifying good grasping poses with Bayesian inference becomes difficult due to two reasons: i) generating data from uninformative priors proves to be inefficient, and ii) the posterior often entails a complex distribution defined on a Riemannian manifold. In this study, we explore the use of implicit representations to construct scene-dependent priors, thereby enabling the application of efficient simulation-based Bayesian inference algorithms for determining successful grasp poses in unstructured environments. Results from both simulation and physical benchmarks showcase the high success rate and promising potential of this approach.

Matthias Pirlet, Adrien Bolland, Alexandre Huynen et al.

Estimating generation costs from observed electricity market data is essential for market simulation, strategic bidding, and system planning. To that end, we model the relationship between generation costs and production schedules with a latent variable model. Estimating generation costs from observed schedules is then formulated as Bayesian inference. A prior distribution encodes an initial belief on parameters, and the inference consists of updating the belief with the posterior distribution given observations. We use balanced neural posterior estimation (BNPE) to learn this posterior. Validation on the IEEE RTS-96 test system shows that marginal costs are recovered with narrow credible intervals, while start-up costs remain largely unidentifiable from schedules alone. The method is benchmarked against an inverse-optimization algorithm that exhibits larger parameter errors without uncertainty quantification.

Franciszek Szewczyk, Gilles Louppe, Matthia Sabatelli

Lifelike visualizations in design, cinematography, and gaming rely on precise physics simulations, typically requiring extensive computational resources and detailed physical input. This paper presents a method that can infer a system's physical properties from a short video, eliminating the need for explicit parameter input, provided it is close to the training condition. The learned representation is then used within a Graph Network-based Simulator to emulate the trajectories of physical systems. We demonstrate that the video-derived encodings effectively capture the physical properties of the system and showcase a linear dependence between some of the encodings and the system's motion.

Omer Rochman Sharabi, Sacha Lewin, Gilles Louppe

Mesh-free Lagrangian methods are widely used for simulating fluids, solids, and their complex interactions due to their ability to handle large deformations and topological changes. These physics simulators, however, require substantial computational resources for accurate simulations. To address these issues, deep learning emulators promise faster and scalable simulations, yet they often remain expensive and difficult to train, limiting their practical use. Inspired by the Material Point Method (MPM), we present NeuralMPM, a neural emulation framework for particle-based simulations. NeuralMPM interpolates Lagrangian particles onto a fixed-size grid, computes updates on grid nodes using image-to-image neural networks, and interpolates back to the particles. Similarly to MPM, NeuralMPM benefits from the regular voxelized representation to simplify the computation of the state dynamics, while avoiding the drawbacks of mesh-based Eulerian methods. We demonstrate the advantages of NeuralMPM on several datasets, including fluid dynamics and fluid-solid interactions. Compared to existing methods, NeuralMPM reduces training times from days to hours, while achieving comparable or superior long-term accuracy, making it a promising approach for practical forward and inverse problems. A project page is available at https://neuralmpm.isach.be

Benjamin Kurt Miller, Alex Cole, Gilles Louppe et al.

We present algorithms (a) for nested neural likelihood-to-evidence ratio estimation, and (b) for simulation reuse via an inhomogeneous Poisson point process cache of parameters and corresponding simulations. Together, these algorithms enable automatic and extremely simulator efficient estimation of marginal and joint posteriors. The algorithms are applicable to a wide range of physics and astronomy problems and typically offer an order of magnitude better simulator efficiency than traditional likelihood-based sampling methods. Our approach is an example of likelihood-free inference, thus it is also applicable to simulators which do not offer a tractable likelihood function. Simulator runs are never rejected and can be automatically reused in future analysis. As functional prototype implementation we provide the open-source software package swyft.

Antoine Wehenkel, Gilles Louppe

Normalizing flows model complex probability distributions by combining a base distribution with a series of bijective neural networks. State-of-the-art architectures rely on coupling and autoregressive transformations to lift up invertible functions from scalars to vectors. In this work, we revisit these transformations as probabilistic graphical models, showing they reduce to Bayesian networks with a pre-defined topology and a learnable density at each node. From this new perspective, we propose the graphical normalizing flow, a new invertible transformation with either a prescribed or a learnable graphical structure. This model provides a promising way to inject domain knowledge into normalizing flows while preserving both the interpretability of Bayesian networks and the representation capacity of normalizing flows. We show that graphical conditioners discover relevant graph structure when we cannot hypothesize it. In addition, we analyze the effect of $\ell_1$-penalization on the recovered structure and on the quality of the resulting density estimation. Finally, we show that graphical conditioners lead to competitive white box density estimators. Our implementation is available at https://github.com/AWehenkel/DAG-NF.

Matthias Pirlet, Adrien Bolland, Gilles Louppe et al.

The Unit Commitment (UC) problem is a key optimization task in power systems to forecast the generation schedules of power units over a finite time period by minimizing costs while meeting demand and technical constraints. However, many parameters required by the UC problem are unknown, such as the costs. In this work, we estimate these unknown costs using simulation-based inference on an illustrative UC problem, which provides an approximated posterior distribution of the parameters given observed generation schedules and demands. Our results highlight that the learned posterior distribution effectively captures the underlying distribution of the data, providing a range of possible values for the unknown parameters given a past observation. This posterior allows for the estimation of past costs using observed past generation schedules, enabling operators to better forecast future costs and make more robust generation scheduling forecasts. We present avenues for future research to address overconfidence in posterior estimation, enhance the scalability of the methodology and apply it to more complex UC problems modeling the network constraints and renewable energy sources.

François Rozet, Gérôme Andry, François Lanusse et al.

Diffusion models recently proved to be remarkable priors for Bayesian inverse problems. However, training these models typically requires access to large amounts of clean data, which could prove difficult in some settings. In this work, we present DiEM, a novel method based on the expectation-maximization algorithm for training diffusion models from incomplete and noisy observations only. Unlike previous works, DiEM leads to proper diffusion models, which is crucial for downstream tasks. As part of our methods, we propose and motivate an improved posterior sampling scheme for unconditional diffusion models. We present empirical evidence supporting the effectiveness of our approach.

Gérôme Andry, Sacha Lewin, François Rozet et al.

Deep learning has advanced weather forecasting, but accurate predictions first require identifying the current state of the atmosphere from observational data. In this work, we introduce Appa, a score-based data assimilation model generating global atmospheric trajectories at 0.25\si{\degree} resolution and 1-hour intervals. Powered by a 565M-parameter latent diffusion model trained on ERA5, Appa can be conditioned on arbitrary observations to infer plausible trajectories, without retraining. Our probabilistic framework handles reanalysis, filtering, and forecasting, within a single model, producing physically consistent reconstructions from various inputs. Results establish latent score-based data assimilation as a promising foundation for future global atmospheric modeling systems.

François Rozet, Ruben Ohana, Michael McCabe et al.

The steep computational cost of diffusion models at inference hinders their use as fast physics emulators. In the context of image and video generation, this computational drawback has been addressed by generating in the latent space of an autoencoder instead of the pixel space. In this work, we investigate whether a similar strategy can be effectively applied to the emulation of dynamical systems and at what cost. We find that the accuracy of latent-space emulation is surprisingly robust to a wide range of compression rates (up to 1000x). We also show that diffusion-based emulators are consistently more accurate than non-generative counterparts and compensate for uncertainty in their predictions with greater diversity. Finally, we cover practical design choices, spanning from architectures to optimizers, that we found critical to train latent-space emulators.

Omer Rochman, Gilles Louppe

Neural PDE solvers used for scientific simulation often violate governing equation constraints. While linear constraints can be projected cheaply, many constraints are nonlinear, complicating projection onto the feasible set. Dynamical PDEs are especially difficult because constraints induce long-range dependencies in time. In this work, we evaluate two training-free, post hoc projections of approximate solutions: a nonlinear optimization-based projection, and a local linearization-based projection using Jacobian-vector and vector-Jacobian products. We analyze constraints across representative PDEs and find that both projections substantially reduce violations and improve accuracy over physics-informed baselines.

Thomas Savary, François Rozet, Gilles Louppe

Data assimilation is widely used in many disciplines such as meteorology, oceanography, and robotics to estimate the state of a dynamical system from noisy observations. In this work, we propose a lightweight and general method to perform data assimilation using diffusion models pre-trained for emulating dynamical systems. Our method builds on particle filters, a class of data assimilation algorithms, and does not require any further training. As a guiding example throughout this work, we illustrate our methodology on GenCast, a diffusion-based model that generates global ensemble weather forecasts.

Adrien Bolland, Gilles Louppe, Damien Ernst

Direct policy optimization in reinforcement learning is usually solved with policy-gradient algorithms, which optimize policy parameters via stochastic gradient ascent. This paper provides a new theoretical interpretation and justification of these algorithms. First, we formulate direct policy optimization in the optimization by continuation framework. The latter is a framework for optimizing nonconvex functions where a sequence of surrogate objective functions, called continuations, are locally optimized. Second, we show that optimizing affine Gaussian policies and performing entropy regularization can be interpreted as implicitly optimizing deterministic policies by continuation. Based on these theoretical results, we argue that exploration in policy-gradient algorithms consists in computing a continuation of the return of the policy at hand, and that the variance of policies should be history-dependent functions adapted to avoid local extrema rather than to maximize the return of the policy.

Antoine Wehenkel, Jens Behrmann, Hsiang Hsu et al.

Hybrid modelling reduces the misspecification of expert models by combining them with machine learning (ML) components learned from data. Similarly to many ML algorithms, hybrid model performance guarantees are limited to the training distribution. Leveraging the insight that the expert model is usually valid even outside the training domain, we overcome this limitation by introducing a hybrid data augmentation strategy termed \textit{expert augmentation}. Based on a probabilistic formalization of hybrid modelling, we demonstrate that expert augmentation, which can be incorporated into existing hybrid systems, improves generalization. We empirically validate the expert augmentation on three controlled experiments modelling dynamical systems with ordinary and partial differential equations. Finally, we assess the potential real-world applicability of expert augmentation on a dataset of a real double pendulum.

Arnaud Delaunoy, Gilles Louppe

Computing the Bayesian posterior of a neural network is a challenging task due to the high-dimensionality of the parameter space. Anchored ensembles approximate the posterior by training an ensemble of neural networks on anchored losses designed for the optima to follow the Bayesian posterior. Training an ensemble, however, becomes computationally expensive as its number of members grows since the full training procedure is repeated for each member. In this note, we present Sequential Anchored Ensembles (SAE), a lightweight alternative to anchored ensembles. Instead of training each member of the ensemble from scratch, the members are trained sequentially on losses sampled with high auto-correlation, hence enabling fast convergence of the neural networks and efficient approximation of the Bayesian posterior. SAE outperform anchored ensembles, for a given computational budget, on some benchmarks while showing comparable performance on the others and achieved 2nd and 3rd place in the light and extended tracks of the NeurIPS 2021 Approximate Inference in Bayesian Deep Learning competition.

Antonio Sutera, Gilles Louppe, Van Anh Huynh-Thu et al.

Random forests have been widely used for their ability to provide so-called importance measures, which give insight at a global (per dataset) level on the relevance of input variables to predict a certain output. On the other hand, methods based on Shapley values have been introduced to refine the analysis of feature relevance in tree-based models to a local (per instance) level. In this context, we first show that the global Mean Decrease of Impurity (MDI) variable importance scores correspond to Shapley values under some conditions. Then, we derive a local MDI importance measure of variable relevance, which has a very natural connection with the global MDI measure and can be related to a new notion of local feature relevance. We further link local MDI importances with Shapley values and discuss them in the light of related measures from the literature. The measures are illustrated through experiments on several classification and regression problems.

Joeri Hermans, Arnaud Delaunoy, François Rozet et al.

We present extensive empirical evidence showing that current Bayesian simulation-based inference algorithms can produce computationally unfaithful posterior approximations. Our results show that all benchmarked algorithms -- (Sequential) Neural Posterior Estimation, (Sequential) Neural Ratio Estimation, Sequential Neural Likelihood and variants of Approximate Bayesian Computation -- can yield overconfident posterior approximations, which makes them unreliable for scientific use cases and falsificationist inquiry. Failing to address this issue may reduce the range of applicability of simulation-based inference. For this reason, we argue that research efforts should be made towards theoretical and methodological developments of conservative approximate inference algorithms and present research directions towards this objective. In this regard, we show empirical evidence that ensembling posterior surrogates provides more reliable approximations and mitigates the issue.

François Rozet, Gilles Louppe

In many areas of science, complex phenomena are modeled by stochastic parametric simulators, often featuring high-dimensional parameter spaces and intractable likelihoods. In this context, performing Bayesian inference can be challenging. In this work, we present a novel method that enables amortized inference over arbitrary subsets of the parameters, without resorting to numerical integration, which makes interpretation of the posterior more convenient. Our method is efficient and can be implemented with arbitrary neural network architectures. We demonstrate the applicability of the method on parameter inference of binary black hole systems from gravitational waves observations.

Norman Marlier, Olivier Brüls, Gilles Louppe

Multi-fingered robotic grasping is an undeniable stepping stone to universal picking and dexterous manipulation. Yet, multi-fingered grippers remain challenging to control because of their rich nonsmooth contact dynamics or because of sensor noise. In this work, we aim to plan hand configurations by performing Bayesian posterior inference through the full stochastic forward simulation of the robot in its environment, hence robustly accounting for many of the uncertainties in the system. While previous methods either relied on simplified surrogates of the likelihood function or attempted to learn to directly predict maximum likelihood estimates, we bring a novel simulation-based approach for full Bayesian inference based on a deep neural network surrogate of the likelihood-to-evidence ratio. Hand configurations are found by directly optimizing through the resulting amortized and differentiable expression for the posterior. The geometry of the configuration space is accounted for by proposing a Riemannian manifold optimization procedure through the neural posterior. Simulation and physical benchmarks demonstrate the high success rate of the procedure.

Benjamin Kurt Miller, Alex Cole, Patrick Forré et al.

Parametric stochastic simulators are ubiquitous in science, often featuring high-dimensional input parameters and/or an intractable likelihood. Performing Bayesian parameter inference in this context can be challenging. We present a neural simulation-based inference algorithm which simultaneously offers simulation efficiency and fast empirical posterior testability, which is unique among modern algorithms. Our approach is simulation efficient by simultaneously estimating low-dimensional marginal posteriors instead of the joint posterior and by proposing simulations targeted to an observation of interest via a prior suitably truncated by an indicator function. Furthermore, by estimating a locally amortized posterior our algorithm enables efficient empirical tests of the robustness of the inference results. Since scientists cannot access the ground truth, these tests are necessary for trusting inference in real-world applications. We perform experiments on a marginalized version of the simulation-based inference benchmark and two complex and narrow posteriors, highlighting the simulator efficiency of our algorithm as well as the quality of the estimated marginal posteriors.

Antoine Wehenkel, Gilles Louppe

Among likelihood-based approaches for deep generative modelling, variational autoencoders (VAEs) offer scalable amortized posterior inference and fast sampling. However, VAEs are also more and more outperformed by competing models such as normalizing flows (NFs), deep-energy models, or the new denoising diffusion probabilistic models (DDPMs). In this preliminary work, we improve VAEs by demonstrating how DDPMs can be used for modelling the prior distribution of the latent variables. The diffusion prior model improves upon Gaussian priors of classical VAEs and is competitive with NF-based priors. Finally, we hypothesize that hierarchical VAEs could similarly benefit from the enhanced capacity of diffusion priors.

Thibaut Théate, Antoine Wehenkel, Adrien Bolland et al.

The distributional reinforcement learning (RL) approach advocates for representing the complete probability distribution of the random return instead of only modelling its expectation. A distributional RL algorithm may be characterised by two main components, namely the representation of the distribution together with its parameterisation and the probability metric defining the loss. The present research work considers the unconstrained monotonic neural network (UMNN) architecture, a universal approximator of continuous monotonic functions which is particularly well suited for modelling different representations of a distribution. This property enables the efficient decoupling of the effect of the function approximator class from that of the probability metric. The research paper firstly introduces a methodology for learning different representations of the random return distribution (PDF, CDF and QF). Secondly, a novel distributional RL algorithm named unconstrained monotonic deep Q-network (UMDQN) is presented. To the authors' knowledge, it is the first distributional RL method supporting the learning of three, valid and continuous representations of the random return distribution. Lastly, in light of this new algorithm, an empirical comparison is performed between three probability quasi-metrics, namely the Kullback-Leibler divergence, Cramer distance, and Wasserstein distance. The results highlight the main strengths and weaknesses associated with each probability metric together with an important limitation of the Wasserstein distance.

Pedro L. C. Rodrigues, Thomas Moreau, Gilles Louppe et al.

Inferring the parameters of a stochastic model based on experimental observations is central to the scientific method. A particularly challenging setting is when the model is strongly indeterminate, i.e. when distinct sets of parameters yield identical observations. This arises in many practical situations, such as when inferring the distance and power of a radio source (is the source close and weak or far and strong?) or when estimating the amplifier gain and underlying brain activity of an electrophysiological experiment. In this work, we present hierarchical neural posterior estimation (HNPE), a novel method for cracking such indeterminacy by exploiting additional information conveyed by an auxiliary set of observations sharing global parameters. Our method extends recent developments in simulation-based inference (SBI) based on normalizing flows to Bayesian hierarchical models. We validate quantitatively our proposal on a motivating example amenable to analytical solutions and then apply it to invert a well known non-linear model from computational neuroscience, using both simulated and real EEG data.

Pascal Leroy, Damien Ernst, Pierre Geurts et al.

This paper introduces four new algorithms that can be used for tackling multi-agent reinforcement learning (MARL) problems occurring in cooperative settings. All algorithms are based on the Deep Quality-Value (DQV) family of algorithms, a set of techniques that have proven to be successful when dealing with single-agent reinforcement learning problems (SARL). The key idea of DQV algorithms is to jointly learn an approximation of the state-value function $V$, alongside an approximation of the state-action value function $Q$. We follow this principle and generalise these algorithms by introducing two fully decentralised MARL algorithms (IQV and IQV-Max) and two algorithms that are based on the centralised training with decentralised execution training paradigm (QVMix and QVMix-Max). We compare our algorithms with state-of-the-art MARL techniques on the popular StarCraft Multi-Agent Challenge (SMAC) environment. We show competitive results when QVMix and QVMix-Max are compared to well-known MARL techniques such as QMIX and MAVEN and show that QVMix can even outperform them on some of the tested environments, being the algorithm which performs best overall. We hypothesise that this is due to the fact that QVMix suffers less from the overestimation bias of the $Q$ function.

Joeri Hermans, Nilanjan Banik, Christoph Weniger et al.

A statistical analysis of the observed perturbations in the density of stellar streams can in principle set stringent contraints on the mass function of dark matter subhaloes, which in turn can be used to constrain the mass of the dark matter particle. However, the likelihood of a stellar density with respect to the stream and subhaloes parameters involves solving an intractable inverse problem which rests on the integration of all possible forward realisations implicitly defined by the simulation model. In order to infer the subhalo abundance, previous analyses have relied on Approximate Bayesian Computation (ABC) together with domain-motivated but handcrafted summary statistics. Here, we introduce a likelihood-free Bayesian inference pipeline based on Amortised Approximate Likelihood Ratios (AALR), which automatically learns a mapping between the data and the simulator parameters and obviates the need to handcraft a possibly insufficient summary statistic. We apply the method to the simplified case where stellar streams are only perturbed by dark matter subhaloes, thus neglecting baryonic substructures, and describe several diagnostics that demonstrate the effectiveness of the new method and the statistical quality of the learned estimator.

Maxime Vandegar, Michael Kagan, Antoine Wehenkel et al.

We revisit empirical Bayes in the absence of a tractable likelihood function, as is typical in scientific domains relying on computer simulations. We investigate how the empirical Bayesian can make use of neural density estimators first to use all noise-corrupted observations to estimate a prior or source distribution over uncorrupted samples, and then to perform single-observation posterior inference using the fitted source distribution. We propose an approach based on the direct maximization of the log-marginal likelihood of the observations, examining both biased and de-biased estimators, and comparing to variational approaches. We find that, up to symmetries, a neural empirical Bayes approach recovers ground truth source distributions. With the learned source distribution in hand, we show the applicability to likelihood-free inference and examine the quality of the resulting posterior estimates. Finally, we demonstrate the applicability of Neural Empirical Bayes on an inverse problem from collider physics.

Arnaud Delaunoy, Antoine Wehenkel, Tanja Hinderer et al.

Gravitational waves from compact binaries measured by the LIGO and Virgo detectors are routinely analyzed using Markov Chain Monte Carlo sampling algorithms. Because the evaluation of the likelihood function requires evaluating millions of waveform models that link between signal shapes and the source parameters, running Markov chains until convergence is typically expensive and requires days of computation. In this extended abstract, we provide a proof of concept that demonstrates how the latest advances in neural simulation-based inference can speed up the inference time by up to three orders of magnitude -- from days to minutes -- without impairing the performance. Our approach is based on a convolutional neural network modeling the likelihood-to-evidence ratio and entirely amortizes the computation of the posterior. We find that our model correctly estimates credible intervals for the parameters of simulated gravitational waves.

Antoine Wehenkel, Gilles Louppe

Normalizing flows have emerged as an important family of deep neural networks for modelling complex probability distributions. In this note, we revisit their coupling and autoregressive transformation layers as probabilistic graphical models and show that they reduce to Bayesian networks with a pre-defined topology and a learnable density at each node. From this new perspective, we provide three results. First, we show that stacking multiple transformations in a normalizing flow relaxes independence assumptions and entangles the model distribution. Second, we show that a fundamental leap of capacity emerges when the depth of affine flows exceeds 3 transformation layers. Third, we prove the non-universality of the affine normalizing flow, regardless of its depth.

Kyle Cranmer, Johann Brehmer, Gilles Louppe

Many domains of science have developed complex simulations to describe phenomena of interest. While these simulations provide high-fidelity models, they are poorly suited for inference and lead to challenging inverse problems. We review the rapidly developing field of simulation-based inference and identify the forces giving new momentum to the field. Finally, we describe how the frontier is expanding so that a broad audience can appreciate the profound change these developments may have on science.

Johann Brehmer, Siddharth Mishra-Sharma, Joeri Hermans et al.

The subtle and unique imprint of dark matter substructure on extended arcs in strong lensing systems contains a wealth of information about the properties and distribution of dark matter on small scales and, consequently, about the underlying particle physics. However, teasing out this effect poses a significant challenge since the likelihood function for realistic simulations of population-level parameters is intractable. We apply recently-developed simulation-based inference techniques to the problem of substructure inference in galaxy-galaxy strong lenses. By leveraging additional information extracted from the simulator, neural networks are efficiently trained to estimate likelihood ratios associated with population-level parameters characterizing substructure. Through proof-of-principle application to simulated data, we show that these methods can provide an efficient and principled way to simultaneously analyze an ensemble of strong lenses, and can be used to mine the large sample of lensing images deliverable by near-future surveys for signatures of dark matter substructure.

Matthia Sabatelli, Gilles Louppe, Pierre Geurts et al.

This paper makes one step forward towards characterizing a new family of \textit{model-free} Deep Reinforcement Learning (DRL) algorithms. The aim of these algorithms is to jointly learn an approximation of the state-value function ($V$), alongside an approximation of the state-action value function ($Q$). Our analysis starts with a thorough study of the Deep Quality-Value Learning (DQV) algorithm, a DRL algorithm which has been shown to outperform popular techniques such as Deep-Q-Learning (DQN) and Double-Deep-Q-Learning (DDQN) \cite{sabatelli2018deep}. Intending to investigate why DQV's learning dynamics allow this algorithm to perform so well, we formulate a set of research questions which help us characterize a new family of DRL algorithms. Among our results, we present some specific cases in which DQV's performance can get harmed and introduce a novel \textit{off-policy} DRL algorithm, called DQV-Max, which can outperform DQV. We then study the behavior of the $V$ and $Q$ functions that are learned by DQV and DQV-Max and show that both algorithms might perform so well on several DRL test-beds because they are less prone to suffer from the overestimation bias of the $Q$ function.

Antoine Wehenkel, Gilles Louppe

Monotonic neural networks have recently been proposed as a way to define invertible transformations. These transformations can be combined into powerful autoregressive flows that have been shown to be universal approximators of continuous probability distributions. Architectures that ensure monotonicity typically enforce constraints on weights and activation functions, which enables invertibility but leads to a cap on the expressiveness of the resulting transformations. In this work, we propose the Unconstrained Monotonic Neural Network (UMNN) architecture based on the insight that a function is monotonic as long as its derivative is strictly positive. In particular, this latter condition can be enforced with a free-form neural network whose only constraint is the positiveness of its output. We evaluate our new invertible building block within a new autoregressive flow (UMNN-MAF) and demonstrate its effectiveness on density estimation experiments. We also illustrate the ability of UMNNs to improve variational inference.

Johann Brehmer, Kyle Cranmer, Irina Espejo et al.

One major challenge for the legacy measurements at the LHC is that the likelihood function is not tractable when the collected data is high-dimensional and the detector response has to be modeled. We review how different analysis strategies solve this issue, including the traditional histogram approach used in most particle physics analyses, the Matrix Element Method, Optimal Observables, and modern techniques based on neural density estimation. We then discuss powerful new inference methods that use a combination of matrix element information and machine learning to accurately estimate the likelihood function. The MadMiner package automates all necessary data-processing steps. In first studies we find that these new techniques have the potential to substantially improve the sensitivity of the LHC legacy measurements.

Joeri Hermans, Volodimir Begy, Gilles Louppe

Posterior inference with an intractable likelihood is becoming an increasingly common task in scientific domains which rely on sophisticated computer simulations. Typically, these forward models do not admit tractable densities forcing practitioners to make use of approximations. This work introduces a novel approach to address the intractability of the likelihood and the marginal model. We achieve this by learning a flexible amortized estimator which approximates the likelihood-to-evidence ratio. We demonstrate that the learned ratio estimator can be embedded in MCMC samplers to approximate likelihood-ratios between consecutive states in the Markov chain, allowing us to draw samples from the intractable posterior. Techniques are presented to improve the numerical stability and to measure the quality of an approximation. The accuracy of our approach is demonstrated on a variety of benchmarks against well-established techniques. Scientific applications in physics show its applicability.

Arthur Pesah, Antoine Wehenkel, Gilles Louppe

Likelihood-free inference is concerned with the estimation of the parameters of a non-differentiable stochastic simulator that best reproduce real observations. In the absence of a likelihood function, most of the existing inference methods optimize the simulator parameters through a handcrafted iterative procedure that tries to make the simulated data more similar to the observations. In this work, we explore whether meta-learning can be used in the likelihood-free context, for learning automatically from data an iterative optimization procedure that would solve likelihood-free inference problems. We design a recurrent inference machine that learns a sequence of parameter updates leading to good parameter estimates, without ever specifying some explicit notion of divergence between the simulated data and the real data distributions. We demonstrate our approach on toy simulators, showing promising results both in terms of performance and robustness.

Matthia Sabatelli, Gilles Louppe, Pierre Geurts et al.

We introduce a novel Deep Reinforcement Learning (DRL) algorithm called Deep Quality-Value (DQV) Learning. DQV uses temporal-difference learning to train a Value neural network and uses this network for training a second Quality-value network that learns to estimate state-action values. We first test DQV's update rules with Multilayer Perceptrons as function approximators on two classic RL problems, and then extend DQV with the use of Deep Convolutional Neural Networks, `Experience Replay' and `Target Neural Networks' for tackling four games of the Atari Arcade Learning environment. Our results show that DQV learns significantly faster and better than Deep Q-Learning and Double Deep Q-Learning, suggesting that our algorithm can potentially be a better performing synchronous temporal difference algorithm than what is currently present in DRL.

Markus Stoye, Johann Brehmer, Gilles Louppe et al.

We extend recent work (Brehmer, et. al., 2018) that use neural networks as surrogate models for likelihood-free inference. As in the previous work, we exploit the fact that the joint likelihood ratio and joint score, conditioned on both observed and latent variables, can often be extracted from an implicit generative model or simulator to augment the training data for these surrogate models. We show how this augmented training data can be used to provide a new cross-entropy estimator, which provides improved sample efficiency compared to previous loss functions exploiting this augmented training data.