Parameter Estimation

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Overview

Parameter estimation uses observed data to infer the unknown settings that govern a model, enabling prediction, scientific explanation, and control. Approaches range from maximum likelihood and Bayesian methods to simulation-based inference, gradient or evolutionary optimization, and physics-informed neural networks that embed known equations. Researchers apply these tools to noisy, incomplete observations in settings such as dynamical systems, medical models, imaging, and scientific simulations, often estimating uncertainty as well as a best-fit value. A central challenge is that good predictions do not guarantee that parameters are uniquely or accurately identified, especially with sparse data, model mismatch, or costly simulations.

Latest papers

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144 papers20 papers in the last four weeks, up 67% on the four weeks before. 0.1% of all new papers.

Mar 19, 2026

Enhancing the Parameterization of Reservoir Properties for Data Assimilation Using Deep VAE-GAN

Currently, the methods called Iterative Ensemble Smoothers, especially the method called Ensemble Smoother with Multiple Data Assimilation (ESMDA) can be considered state-of-the-art for history matching in petroleum reservoir simulation. However, this approach has two important limitations: the use of an ensemble with finite size to represent the distributions and the Gaussian assumption in parameter and data uncertainties. This latter is particularly important because many reservoir properties have non-Gaussian distributions. Parameterization involves mapping non-Gaussian parameters to a Gaussian field before the update and then mapping them back to the original domain to forward the ensemble through the reservoir simulator. A promising approach to perform parameterization is through deep learning models. Recent studies have shown that Generative Adversarial Networks (GAN) performed poorly concerning data assimilation, but generated more geologically plausible realizations of the reservoir, while the Variational Autoencoder (VAE) performed better than the GAN in data assimilation, but generated less geologically realistic models. This work is innovative in combining the strengths of both to implement a deep learning model called Variational Autoencoder Generative Adversarial Network (VAE-GAN) integrated with ESMDA. The methodology was applied in two case studies, one case being categorical and the other with continuous values of permeability. Our findings demonstrate that by applying the VAE-GAN model we can obtain high quality reservoir descriptions (just like GANs) and a good history matching on the production curves (just like VAEs) simultaneously.

M. A. Sampaio, P. H. Ranazzi, M. J. Blunt

Departamento de Engenharia de Minas e de Petróleo, Escola Politécnica, Universidade de São Paulo, 05508-030, SP, Brasil. · Department of Earth Science and Engineering, Imperial College London, South Kensington, London, UK

Data AssimilationParameter Estimation
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Mar 16, 2026

INSTRUMENTAL: Automatic Synthesizer Parameter Recovery from Audio via Evolutionary Optimization

Existing audio-to-MIDI tools extract notes but discard the timbral characteristics that define an instrument's identity. We present Instrumental, a system that recovers continuous synthesizer parameters from audio by coupling a differentiable 28-parameter subtractive synthesizer with CMA-ES, a derivative-free evolutionary optimizer. We optimize a composite perceptual loss combining mel-scaled STFT, spectral centroid, and MFCC divergence, achieving a matching loss of 2.09 on real recorded audio. We systematically evaluate eight hypotheses for improving convergence and find that only parametric EQ boosting yields meaningful improvement. Our results show that CMA-ES outperforms gradient descent on this non-convex landscape, that more parameters do not monotonically improve matching, and that spectral analysis initialization accelerates convergence over random starts.

Philipp Bogdan

Parameter EstimationEvolutionary Optimization
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Mar 16, 2026

UNICORN: Ultrasound Nakagami Imaging via Score Matching and Adaptation for Assessing Hepatic Steatosis

Ultrasound imaging is an essential first-line tool for assessing hepatic steatosis. While conventional B-mode ultrasound imaging has limitations in providing detailed tissue characterization, ultrasound Nakagami imaging holds promise for visualizing and quantifying tissue scattering in backscattered signals, with potential applications in fat fraction analysis. However, existing methods for Nakagami imaging struggle with optimal window size selection and suffer from estimator instability, leading to degraded image resolution. To address these challenges, we propose a novel method called UNICORN (Ultrasound Nakagami Imaging via Score Matching and Adaptation), which offers an accurate, closed-form estimator for Nakagami parameter estimation based on the score function of the ultrasound envelope signal. Unlike methods that visualize only specific regions of interest (ROI) and estimate parameters within fixed window sizes, our approach provides comprehensive parameter mapping by providing a pixel-by-pixel estimator, resulting in high-resolution imaging. We demonstrated that our proposed estimator effectively assesses hepatic steatosis and provides visual distinction in the backscattered statistics associated with this condition. Through extensive experiments using real envelope data from patient, we validated that UNICORN enables clinical detection of hepatic steatosis and exhibits robustness and generalizability.

Kwanyoung Kim, Jaa-Yeon Lee, Youngjun Ko +2

Department of AI Convergence, GIST · Kim Jae Chul Graduate School of AI, KAIST · Samsung Medison

Ultrasound ImagingParameter Estimation
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Mar 12, 2026

Inverse Neural Operator for ODE Parameter Optimization

We propose the Inverse Neural Operator (INO), a two-stage framework for recovering hidden ODE parameters from sparse, partial observations. In Stage 1, a Conditional Fourier Neural Operator (C-FNO) with cross-attention learns a differentiable surrogate that reconstructs full ODE trajectories from arbitrary sparse inputs, suppressing high-frequency artifacts via spectral regularization. In Stage 2, an Amortized Drifting Model (ADM) learns a kernel-weighted velocity field in parameter space, transporting random parameter initializations toward the ground truth without backpropagating through the surrogate, avoiding the Jacobian instabilities that afflict gradient-based inversion in stiff regimes. Experiments on a real-world stiff atmospheric chemistry benchmark (POLLU, 25 parameters) and a synthetic Gene Regulatory Network (GRN, 40 parameters) show that INO outperforms gradient-based and amortized baselines in parameter recovery accuracy while requiring only 0.23s inference time, a 487x speedup over iterative gradient descent.

Zhi-Song Liu, Wenqing Peng, Helmi Toropainen +5

Department of Computational Engineering, Lappeenranta-Lahti University of Technology, Lahti, Finland · Atmospheric Modelling Center Lahti (AMC-Lahti), Finland · University of Helsinki, Helsinki, Finland +3

Parameter EstimationOrdinary Differential EquationsNeural OperatorsLearning to OptimizeNeural Surrogate Modeling
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Mar 10, 2026

First Estimation of Model Parameters for Neutrino-Induced Nucleon Knockout Using Simulation-Based Inference

To enable an accurate determination of oscillation parameters, accelerator-based neutrino experiments require detailed simulations of nuclear interaction physics in the GeV regime. While substantial effort from both theory and experiment is currently being invested to improve the fidelity of these simulations, their present deficiencies typically oblige experimental collaborations to resort to empirical tuning of simulation model parameters. As the precision requirements of the field continue to become more stringent, machine learning techniques may provide a powerful means of handling corresponding growth in the complexity of future neutrino interaction model tuning exercises. To study the suitability of simulation-based inference (SBI) for this physics application, in this paper we revisit a tuned configuration of the GENIE neutrino event generator that was originally developed by the MicroBooNE collaboration. Despite closely reproducing the adopted values of four physics parameters when confronted with the tuned cross-section predictions as input, we find that our trained SBI algorithm prefers modestly different values (within MicroBooNE's assigned uncertainties) and achieves slightly better goodness-of-fit when inference is run on the experimental data set originally used by MicroBooNE. We also find that our trained algorithm can create a fair approximation of an alternative neutrino scattering simulation, NuWro, that shares only a subset of its physics model parameters with GENIE.

Karla Tame-Narvaez, Steven Gardiner, Aleksandra Ćiprijanović +1

Fermi National Accelerator Laboratory, Batavia, Illinois 60510 USA · Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637 · NSF-Simons AI Institute for the Sky (SkAI), 172 E. Chestnut St., Chicago, IL 60611

Parameter EstimationModel CalibrationSimulation-Based InferenceHigh-Energy PhysicsPhysics-Based Simulation
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Mar 4, 2026

PDE foundation model-accelerated inverse estimation of system parameters in inertial confinement fusion

PDE foundation models are typically pretrained on large, diverse corpora of PDE datasets and can be adapted to new settings with limited task-specific data. However, most downstream evaluations focus on forward problems, such as autoregressive rollout prediction. In this work, we study an inverse problem in inertial confinement fusion (ICF): estimating system parameters (inputs) from multi-modal, snapshot-style observations (outputs). Using the open JAG benchmark, which provides hyperspectral X-ray images and scalar observables per simulation, we finetune the PDE foundation model and train a lightweight task-specific head to jointly reconstruct hyperspectral images and regress system parameters. The fine-tuned model achieves accurate hyperspectral reconstruction (test MSE 1.2e-3) and strong parameter-estimation performance (up to R^2=0.995). Data-scaling experiments (5%-100% of the training set) show consistent improvements in both reconstruction and regression losses as the amount of training data increases, with the largest marginal gains in the low-data regime. Finally, finetuning from pretrained MORPH weights outperforms training the same architecture from scratch, demonstrating that foundation-model initialization improves sample efficiency for data-limited inverse problems in ICF.

Mahindra Rautela, Alexander Scheinker, Bradley Love +4

AOT-IC Group, Los Alamos National Laboratory, Los Alamos, NM, USA · CAI Division, Los Alamos National Laboratory, Los Alamos, NM, USA · HPC Division, Los Alamos National Laboratory, Los Alamos, NM, USA

PDE Foundation ModelsParameter EstimationPDE Inverse Problems
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Mar 3, 2026

Learning-Augmented Moment Estimation on Time-Decay Models

Motivated by the prevalence and success of machine learning, a line of recent work has studied learning-augmented algorithms in the streaming model. These results have shown that for natural and practical oracles implemented with machine learning models, we can obtain streaming algorithms with improved space efficiency that are otherwise provably impossible. On the other hand, our understanding is much more limited when items are weighted unequally, for example, in the sliding-window model, where older data must be expunged from the dataset, e.g., by privacy regulation laws. In this paper, we utilize an oracle for the heavy-hitters of datasets to give learning-augmented algorithms for a number of fundamental problems, such as norm/moment estimation, frequency estimation, cascaded norms, and rectangular moment estimation, in the time-decay setting. We complement our theoretical results with a number of empirical evaluations that demonstrate the practical efficiency of our algorithms on real and synthetic datasets.

Soham Nagawanshi, Shalini Panthangi, Chen Wang +2

Texas A&M University · Carnegie Mellon University · Rensselaer Polytechnic Institute

Learning-Augmented AlgorithmsStreaming AlgorithmsParameter Estimation
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Mar 2, 2026

Learning graph topology from metapopulation epidemic encoder-decoder

Metapopulation epidemic models are a valuable tool for studying large-scale outbreaks. With the limited availability of epidemic tracing data, it is challenging to infer the essential constituents of these models, namely, the epidemic parameters and the relevant mobility network between subpopulations. Either one of these constituents can be estimated while assuming the other; however, the problem of their joint inference has not yet been solved. Here, we propose two encoder-decoder deep learning architectures that infer metapopulation mobility graphs from time-series data, with and without the assumption of epidemic model parameters. Evaluation across diverse random and empirical mobility networks shows that the proposed approach outperforms the state-of-the-art topology inference. Further, we show that topology inference improves dramatically with data on additional pathogens. Our study establishes a robust framework for simultaneously inferring epidemic parameters and topology, addressing a persistent gap in modeling disease propagation.

Xin Li, Jonathan Cohen, Shai Pilosof +1

Faculty of Computer and Information Science, Ben-Gurion University of the Negev, Be’er Sheva, South District, Israel · Department of Life Sciences, Ben-Gurion University of the Negev, Be’er Sheva, South District, Israel · The Goldman Sonnenfeldt School of Sustainability and Climate Change, Ben-Gurion University of the Negev, Be’er Sheva, South District, Israel

Graph Structure LearningSpatial Epidemic ModelingParameter Estimation
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Mar 2, 2026

Resolving Blind Inverse Problems under Dynamic Range Compression via Structured Forward Operator Modeling

Recovering radiometric fidelity from unknown dynamic range compression (UDRC), such as low-light enhancement and HDR reconstruction, is a challenging blind inverse problem, due to the unknown forward model and irreversible information loss introduced by compression. To address this challenge, we first identify monotonicity as the fundamental physical invariant shared across UDRC tasks. Leveraging this insight, we introduce the cascaded monotonic Bernstein (CaMB) operator to parameterize the unknown forward model. CaMB enforces monotonicity as a hard architectural inductive bias, constraining optimization to physically consistent mappings and enabling robust and stable operator estimation. We further integrate CaMB with a plug-and-play diffusion framework, proposing CaMB-Diff. Within this framework, the diffusion model serves as a powerful geometric prior for structural and semantic recovery, while CaMB explicitly models and corrects radiometric distortions through a physically grounded forward operator. Extensive experiments on a variety of zero-shot UDRC tasks, including low-light enhancement, low-field MRI enhancement, and HDR reconstruction, demonstrate that CaMB-Diff significantly outperforms state-of-the-art zero-shot baselines in terms of both signal fidelity and physical consistency. Moreover, we empirically validate the effectiveness of the proposed CaMB parameterization in accurately modeling the unknown forward operator.

Muyu Liu, Xuanyu Tian, Chenhe Du +3

ShanghaiTech University · Shanghai Jiao Tong University

Image Inverse ProblemsHDR ImagingLow-Light Image EnhancementParameter EstimationDiffusion-Based Inverse Problems
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Feb 27, 2026

Robust Parameter and State Estimation in Multiscale Neuronal Systems Using Physics-Informed Neural Networks

Inferring biophysical parameters and hidden state variables from partial and noisy observations is a fundamental challenge in computational neuroscience. This problem is particularly difficult for fast - slow spiking and bursting models, where strong nonlinearities, multiscale dynamics, and limited observational data often lead to severe sensitivity to initial parameter guesses and convergence failure in the methods replying on the traditional numerical forward solvers. In this work, we developed a physics-informed neural network (PINN) framework for the joint reconstruction of unobserved state variables and the estimation of unknown biophysical parameters in neuronal models. We demonstrate the effectiveness of the method on biophysical neuron models, including the Morris-Lecar model across multiple spiking and bursting regimes and a respiratory model neuron. The method requires only partial voltage observations over short observation windows and remains robust even when initialized with non-informative parameter guesses. These results suggest that PINN can deliver robust and accurate parameter inference and state reconstruction, providing a promising alternative for inverse problems in multiscale neuronal dynamics, where traditional techniques often struggle.

Changliang Wei, Yangyang Wang, Xueyu Zhu

Department of Mathematics, University of Iowa, Iowa City, IA 52246, USA. · Department of Mathematics, Volen National Center for Complex Systems, Brandeis University, Waltham, MA 02453, USA.

Parameter EstimationPhysics-Informed ML
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Feb 20, 2026

PINEAPPLE: Physics-Informed Neuro-Evolution Algorithm for Prognostic Parameter Inference in Lithium-Ion Battery Electrodes

Accurate, real-time, yet non-destructive estimation of internal states in lithium-ion batteries is critical for predicting degradation, optimizing usage strategies, and extending operational lifespan. Here, we introduce PINEAPPLE (Physics-Informed Neuro-Evolution Algorithm for Prognostic Parameter inference in Lithium-ion battery Electrodes), a novel framework that integrates physics-informed neural networks (PINNs) with an evolutionary search algorithm to enable rapid, scalable, and interpretable parameter inference with potential for application to next-generation batteries. The meta-learned PINN utilizes fundamental physics principles to achieve accurate zero-shot prediction of electrode behavior with test errors below 0.1%\% while maintaining an order-of-magnitude speed-up over conventional solvers. PINEAPPLE demonstrates robust parameter inference solely from voltage-time discharge curves across multiple batteries from the open-source CALCE repository, recovering the evolution of key internal state parameters such as Li-ion diffusion coefficients across usage cycles. Notably, the inferred cycle-dependent evolution of these parameters exhibit consistent trends across different batteries without any customized degradation physics-embedded heuristic, highlighting the effective regularizing effect and robustness that can be conferred through incorporation of fundamental physics in PINEAPPLE. By enabling computationally efficient, real-time parameter estimation, PINEAPPLE offers a promising route towards the non-destructive, physics-based characterization of inter-cell and intra-cell variability of battery modules and battery packs, thereby unlocking new opportunities for downstream on-the-fly needs in next-generation battery management systems such as individual cell-scale state-of-health diagnostics.

Karkulali Pugalenthi, Jian Cheng Wong, Qizheng Yang +4

Singapore Institute of Manufacturing Technology, Agency for Science, Technology and Research, 5 Cleantech Loop, Singapore, 636732, Singapore · Institute of High Performance Computing, Agency for Science, Technology and Research, 1 Fusionopolis Way, Singapore, 138632, Singapore · Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR, China +3

Parameter EstimationLithium-Ion BatteriesPhysics-Informed MLNeural Surrogate Modeling
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Feb 19, 2026

Neural Prior Estimation: Learning Class Priors from Latent Representations

Logit adjustment corrects class imbalance using the empirical class prior. We study whether a comparable class-frequency signal can instead be learned from the network representation, without explicitly supplying class counts to the correction rule. We introduce the Neural Prior Estimator (NPE), which attaches one or more lightweight Prior Estimation Modules (PEMs) to the latent representation. Each PEM is trained with a one-way logistic objective on the ground-truth coordinate. The resulting frequency-dependent outputs are combined into an NPE estimate and used as a learned logit correction, forming NPE-LA. In a simplified scalar model, the optimum of the PEM objective is monotone in the class count and grows asymptotically as log⁡Nc\log N_c, up to a slower log⁡log⁡Nc\log \log N_c term. Experiments on long-tailed CIFAR-10 and CIFAR-100 show that NPE-LA is competitive with standard logit adjustment and improves minority-class performance over CE and cRT in the reported settings. Experiments on STARE and ADE20K further show that the same idea can be used as a lightweight recalibration mechanism for dense prediction.

Masoud Yavari, Payman Moallem

Independent Researcher Isfahan, Iran · Department of Electrical Engineering Faculty of Engineering University of Isfahan, Iran

Logit AdjustmentClass-Imbalanced LearningParameter EstimationModel Calibration
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Feb 12, 2026

Estimation of instrument and noise parameters for inverse problem based on prior diffusion model

This article addresses the issue of estimating observation parameters (response and error parameters) in inverse problems. The focus is on cases where regularization is introduced in a Bayesian framework and the prior is modeled by a diffusion process. In this context, the issue of posterior sampling is known to be thorny, and a recent paper proposes a notably simple and effective solution. Additionally, it opens an remarkable flexibility when it comes to estimating observation parameters. The proposed strategy enables to define an optimal estimator for both observation parameters and image of interest. Furthermore, the strategy provides a means for uncertainty quantification. In addition, MCMC algorithms allow for the computation of estimates and properties of posteriors, while offering some guarantees. The paper presents several numerical experiments that clearly confirm the computational efficiency and the quality of both estimates and uncertainty quantification.

Jean-François Giovannelli

Groupe Signal-Image, IMS (Univ. Bordeaux, CNRS, BINP), Talence, France

Parameter EstimationBayesian Inverse ProblemsDiffusion-Based Inverse Problems
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Jan 29, 2026

It's all In the (Exponential) Family: An Equivalence between Maximum Likelihood Estimation and Control Variates for Sketching Algorithms

Maximum likelihood estimators (MLE) and control variate estimators (CVE) have been used in conjunction with known information across sketching algorithms and applications in machine learning. We prove that under certain conditions in an exponential family, an optimal CVE will achieve the same asymptotic variance as the MLE, giving a fixed point algorithm for the MLE. Experiments show the fixed point algorithm is faster and numerically stable compared to other root finding algorithms for the MLE for the bivariate Normal distribution, and we expect this to hold across distributions satisfying these conditions. We show how this algorithm leads to reproducibility for algorithms using MLE / CVE, and demonstrate how the algorithm leads to finding the MLE when the CV weights are known.

Keegan Kang, Kerong Wang, Ding Zhang +3

Department of Mathematics and Statistics, Bucknell University, USA · School of Data Science, University of Virginia, USA · Department of CSE, IIT Hyderabad, India +2

Parameter EstimationExponential FamilyVariance Reduction
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Dec 28, 2025

JADAI: Jointly Amortizing Adaptive Design and Bayesian Inference

We consider problems of parameter estimation where design variables can be actively optimized to maximize information gain. To this end, we introduce JADAI, a framework that jointly amortizes Bayesian adaptive design and inference by training a policy, a history network, and an inference network end-to-end. The networks minimize a generic loss that aggregates incremental reductions in posterior error along experimental sequences. Inference networks are instantiated with diffusion-based posterior estimators that can approximate high-dimensional and multimodal posteriors at every experimental step. Across standard adaptive design benchmarks, JADAI achieves superior or competitive performance.

Niels Bracher, Lars Kühmichel, Desi R. Ivanova +3

1Rensselaer Polytechnic Institute, USA · 2TU Dortmund University, Germany · University of Oxford, UK.

Parameter EstimationAmortized InferenceBayesian InferenceBayesian Experimental Design
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Dec 22, 2025

Diffusion Models in Simulation-Based Inference: A Tutorial Review

Diffusion models have recently emerged as powerful learners for simulation-based inference (SBI), enabling fast and accurate estimation of latent parameters from simulated and real data. Their score-based formulation offers a flexible way to learn conditional or joint distributions over parameters and observations, thereby providing a versatile solution to various modeling problems. In this tutorial review, we synthesize recent developments on diffusion models for SBI, covering design choices for training, inference, and evaluation. We highlight opportunities created by various concepts such as guidance, score composition, flow matching, consistency models, and joint modeling. Furthermore, we discuss how efficiency and statistical accuracy are affected by noise schedules, parameterizations, and samplers. Finally, we illustrate these concepts with case studies across parameter dimensionalities, simulation budgets, and model types, and outline open questions for future research.

Jonas Arruda, Niels Bracher, Ullrich Köthe +2

Bonn Center for Mathematical Life Sciences, Life & Medical Sciences Institute, University of Bonn, Germany · Center for Modeling, Simulation, & Imaging in Medicine (CeMSIM), Rensselaer Polytechnic Institute, NY, USA · Computer Vision and Learning Lab, Heidelberg University, Germany

Simulation-Based InferenceParameter EstimationScore-Based Generative ModelingDiffusion Models
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Dec 5, 2025

Symmetric Linear Dynamical Systems are Learnable from Few Observations

We consider the problem of learning the parameters of a NN-dimensional stochastic linear dynamics under both full and partial observations from a single trajectory of time TT. We introduce and analyze a new estimator that achieves a small maximum element-wise error on the recovery of symmetric dynamic matrices using only T=O(log⁡N)T=\mathcal{O}(\log N) observations, irrespective of whether the matrix is sparse or dense. This estimator is based on the method of moments and does not rely on problem-specific regularization. This is especially important for applications such as structure discovery.

Minh Vu, Andrey Y. Lokhov, Marc Vuffray

Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Dynamical SystemsParameter EstimationStochastic Linear Dynamical SystemsSystem Identification
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Nov 30, 2025

Learning Projection-Aware 360-Degree Image Rectification via Dual-Projection Fusion

Panoramic cameras provide a 360° field of view and are widely used in panoramic vision, immersive visual computing, and robotic perception. However, changes in camera orientation can produce non-upright panoramas, introducing geometric variations that can complicate downstream visual analysis. Existing vision-based rectification methods usually operate within a single projection domain, limiting their ability to jointly exploit local geometric structures and global contextual information. To address this, we formulate 360° image rectification as a projection-aware representation learning problem and propose a dual-projection framework for upright panoramic rectification. A convolutional neural network branch captures local geometric structures from equirectangular projection (ERP) inputs, while a vision transformer branch models global contextual cues from cubemap projections. Cross-projection feature transformation and multi-level feature fusion enable effective interaction between these complementary representations. The learned representation supports collaborative inclination estimation and upright panorama generation, with the two tasks providing complementary geometric and appearance supervision. Experiments on SUN360 and M3D show consistent improvements over existing methods, achieving accuracies within a 1° error threshold of 65.9% and 85.2% and Fréchet Inception Distance scores of 5.87 and 3.26, respectively. Ablation studies verify the contributions of dual-projection representation, cross-projection feature transformation and fusion, and collaborative multi-task learning. The proposed framework provides a projection-aware visual computing approach for panoramic rectification. Code, pretrained models, and training/testing scripts are available at https://github.com/YuhaoShine/DualProjectionFusion.

Yuhao Shan, Qianyi Yuan, Jingguo Liu +3

college of electronic and information engineering, Southwest University, Chongqing 400700, China

Parameter Estimation
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Oct 14, 2025

Learning Latent Energy-Based Models via Interacting Particle Langevin Dynamics

We develop interacting particle algorithms for learning latent variable models with energy-based priors. To do so, we leverage recent developments in particle-based methods for solving maximum marginal likelihood estimation (MMLE) problems. Specifically, we provide a continuous-time framework for learning latent energy-based models, by defining stochastic differential equations (SDEs) that provably solve the MMLE problem. We obtain a practical algorithm as a discretisation of these SDEs and provide theoretical guarantees for the convergence of the proposed algorithm. Finally, we empirically validate the effectiveness of our method on synthetic and image datasets and demonstrate that using a particle based approach offers significant improvement in computational efficiency.

Joanna Marks, Tim Y. J. Wang, O. Deniz Akyildiz

Department of Mathematics, Imperial College London

Parameter EstimationLatent Variable ModelsEnergy-Based ModelsMaximum Marginal Likelihood Estimation
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Oct 2, 2025

Uniform-in-time convergence bounds for Persistent Contrastive Divergence algorithms

We propose a continuous-time formulation of a noisy persistent contrastive divergence (PCD)-like method for maximum likelihood estimation (MLE) of unnormalised densities. Our approach couples parameter updates and sampling of the parametrised density in a multiscale system of stochastic differential equations (SDEs). From this formulation, we derive non-asymptotic bounds for weak test-function errors between the resulting numerical schemes and the MLE point target. The error is decomposed into numerical discretisation, slow-fast averaging, and finite-temperature concentration terms. We also introduce an efficient implementation based on explicit stabilized integrators and establish corresponding long-time error estimates. This leads to a novel method for training energy-based models (EBMs) with quantitative error guarantees.

Paul Felix Valsecchi Oliva, O. Deniz Akyildiz, Andrew Duncan

Department of Mathematics, Imperial College London

Parameter EstimationStochastic ApproximationEnergy-Based ModelsConvergence Analysis
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May 23, 2025

Data-driven multi-agent modelling of calcium interactions in cell culture: PINN vs Regularized Least-squares

Data-driven discovery of dynamics in biological systems allows for better observation and characterization of processes, such as calcium signaling in cell culture. Recent advancements in techniques allow the exploration of previously unattainable insights of dynamical systems, such as the Sparse Identification of Non-Linear Dynamics (SINDy), overcoming the limitations of more classic methodologies. The latter requires some prior knowledge of an effective library of candidate terms, which is not realistic for a real case study. Using inspiration from fields like traffic density estimation and control theory, we propose a methodology for characterization and performance analysis of calcium delivery in a family of cells. In this work, we compare the performance of the Constrained Regularized Least-Squares Method (CRLSM) and Physics-Informed Neural Networks (PINN) for system identification and parameter discovery for governing ordinary differential equations (ODEs). The CRLSM achieves a fairly good parameter estimate and a good data fit when using the learned parameters in the Consensus problem. On the other hand, despite the initial hypothesis, PINNs fail to match the CRLSM performance and, under the current configuration, do not provide fair parameter estimation. However, we have only studied a limited number of PINN architectures, and it is expected that additional hyperparameter tuning, as well as uncertainty quantification, could significantly improve the performance in future works.

Aurora Poggi, Giuseppe Alessio D'Inverno, Hjalmar Brismar +3

KTH Royal Institute of Technology, Department of Mathematics, Stockholm, Sweden · International School for Advanced Studies (SISSA), Trieste, Italy · KTH Royal Institute of Technology, Department of Biophysics, Stockholm, Sweden +1

Nonlinear System IdentificationParameter EstimationPhysics-Informed ML
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Oct 21, 2024

Statistical Inference for Policy Evaluation with Temporal Difference Learning

We investigate the statistical properties of Temporal Difference (TD) learning with Polyak-Ruppert averaging, arguably one of the most widely used algorithms in reinforcement learning, for the task of estimating the parameters of the optimal linear approximation to the value function. Assuming independent samples, we make three theoretical contributions that improve upon the current state-of-the-art results: (i) we establish refined high-dimensional Berry-Esseen bounds over the class of convex sets, achieving faster rates than the best known results, and (ii) we propose and analyze a novel, computationally efficient online plug-in estimator of the asymptotic covariance matrix; (iii) we derive sharper high probability convergence guarantees that depend explicitly on the asymptotic variance and hold under weaker conditions than those adopted in the literature. These results enable the construction of confidence regions and simultaneous confidence intervals for the linear parameters of the value function approximation, with guaranteed finite-sample coverage. We demonstrate the applicability of our theoretical findings through numerical experiments.

Weichen Wu, Gen Li, Yuting Wei +1

The Voleon Group, New York, NY, USA · Department of Statistics, The Chinese University of Hong Kong, Hong Kong SAR. · Department of Statistics and Data Science, The Wharton School, University of Pennsylvania, Philadelphia, PA, USA. +1

Parameter EstimationConfidence Region EstimationValue Function EstimationStochastic ApproximationPolicy EvaluationTemporal-Difference Learning+3
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Oct 1, 2024

FastManly: An EM-Gradient Algorithm for Manly Mixture Models

A faster implementation of mixtures of Manly transformations is proposed. This method, called FastManly, uses Newton's method for optimization in an EM gradient algorithm instead of Nelder-Mead in a traditional EM. A gradient and full Hessian are derived. Simulations show improved performance with noticeable speedups.

Katharine M. Clark, Paul D. McNicholas

Department of Mathematics & Statistics, Trent University, Ontario, Canada. · Department of Mathematics & Statistics, McMaster University, Ontario, Canada.

Parameter EstimationExpectation-Maximization
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Sep 6, 2024

A Deep Learning Framework for TTV Characterization: A Case Study of the Kepler-88 System

Transit timing variation (TTV) provides rich information about the mass and orbital properties of exoplanets, which are often obtained by solving an inverse problem via Markov Chain Monte Carlo (MCMC). In this paper, we design a new data-driven approach, which potentially can be applied to problems that are hard to traditional MCMC methods, such as the case with only one planet transiting. Specifically, we use a deep learning approach to predict the parameters of non-transit companion for the single transit system with transit information (i.e., TTV, and Transit Duration Variation (TDV)) as input. This approach is constructed using the Transformer-based architecture that can extract long-range interactions from TTV sequential data. We focus on two-planet systems based on Kepler-88 as a starting point, with the inner planet fixed to the properties of Kepler-88 and the mass and orbital elements of the outer planet varied. Applying to Kepler-88, we obtained an overall fractional error of ≲\lesssim10% on semi-major axis and eccentricity, and ∼10−30%\sim 10-30\% on planet mass (assuming that the stellar and transiting planet parameters are known), based on a training set of a million simulations. When applied to the test set, our model achieves low fractional errors of less than a few percent for systems with TTV semi-amplitudes ranging from ∼0.3\sim 0.3 s to ∼5\sim 5 min. Extending our approach to a broader range of planetary architectures, including systems with substantially weaker or stronger TTV signals and those containing additional planets or stellar companions, remains an important direction for future work.

Chen Chen, Jiapeng Gao, Gongjie Li +2

School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, GA, 30332, USA · Center for Relativistic Astrophysics, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA · School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA +1

Parameter EstimationAstrophysics
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