Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.
Cancer is a complex disease driven by genomic alterations, and tumor sequencing is becoming a mainstay of clinical care for cancer patients. The emergence of multi-institution sequencing data presents a powerful resource for learning real-world evidence to enhance precision oncology. GENIE BPC, led by the American Association for Cancer Research, establishes a unique database linking genomic data with clinical information for patients treated at multiple cancer centers. However, leveraging such multi-institutional sequencing data presents significant challenges. Variations in gene panels result in loss of information when the analysis is conducted on common gene sets. Additionally, differences in sequencing techniques and patient heterogeneity across institutions add complexity. High data dimensionality, sparse gene mutation patterns, and weak signals at the individual gene level further complicate matters. Motivated by these real-world challenges, we introduce the Bridge model. It uses a quantile-matched latent variable approach to derive integrated features to preserve information beyond common genes and maximize the utilization of all available data while leveraging information sharing to enhance both learning efficiency and the model's capacity to generalize. By extracting harmonized and noise-reduced lower-dimensional latent variables, the true mutation pattern unique to each individual is captured. We assess the model's performance and parameter estimation through extensive simulation studies. The extracted latent features from the Bridge model consistently excel in predicting patient survival across six cancer types in GENIE BPC data.
Dynamic models describe phenomena across scientific disciplines, yet to make these models useful in application the unknown parameter values of the models must be determined. Discrete-time dynamic models are widely used to model biological processes, but it is often difficult to determine these parameters. In this paper, we propose a symbolic-numeric approach for parameter estimation in discrete-time models that involve univariate non-algebraic (locally) analytic functions such as exp. We illustrate the performance (precision) of our approach by applying our approach to two archetypal discrete-time models in biology (the flour beetle 'LPA' model and discrete Lotka-Volterra competition model). Unlike optimization-based methods, our algorithm guarantees to find all solutions of the parameter values up to a specified precision given time-series data for the measured variables provided that there are finitely many parameter values that fit the data and that the used polynomial system solver can find all roots of the associated polynomial system with interval coefficients.
Major depressive disorder (MDD) presents challenges in diagnosis and treatment due to its complex and heterogeneous nature. Emerging evidence indicates that reward processing abnormalities may serve as a behavioral marker for MDD. To measure reward processing, patients perform computer-based behavioral tasks that involve making choices or responding to stimulants that are associated with different outcomes. Reinforcement learning (RL) models are fitted to extract parameters that measure various aspects of reward processing to characterize how patients make decisions in behavioral tasks. Recent findings suggest the inadequacy of characterizing reward learning solely based on a single RL model; instead, there may be a switching of decision-making processes between multiple strategies. An important scientific question is how the dynamics of learning strategies in decision-making affect the reward learning ability of individuals with MDD. Motivated by the probabilistic reward task (PRT) within the EMBARC study, we propose a novel RL-HMM framework for analyzing reward-based decision-making. Our model accommodates learning strategy switching between two distinct approaches under a hidden Markov model (HMM): subjects making decisions based on the RL model or opting for random choices. We account for continuous RL state space and allow time-varying transition probabilities in the HMM. We introduce a computationally efficient EM algorithm for parameter estimation and employ a nonparametric bootstrap for inference. We apply our approach to the EMBARC study to show that MDD patients are less engaged in RL compared to the healthy controls, and engagement is associated with brain activities in the negative affect circuitry during an emotional conflict task.
In neural audio signal processing, pitch conditioning has been used to enhance the performance of synthesizers. However, jointly training pitch estimators and synthesizers is a challenge when using standard audio-to-audio reconstruction loss, leading to reliance on external pitch trackers. To address this issue, we propose using a spectral loss function inspired by optimal transportation theory that minimizes the displacement of spectral energy. We validate this approach through an unsupervised autoencoding task that fits a harmonic template to harmonic signals. We jointly estimate the fundamental frequency and amplitudes of harmonics using a lightweight encoder and reconstruct the signals using a differentiable harmonic synthesizer. The proposed approach offers a promising direction for improving unsupervised parameter estimation in neural audio applications.
In this paper, we propose a novel inverse parameter estimation approach called Bayesian optimized physics-informed neural network (BOPINN). In this study, a PINN solves the partial differential equation (PDE), whereas Bayesian optimization (BO) estimates its parameter. The proposed BOPINN estimates wave velocity associated with wave propagation PDE using a single snapshot observation. An objective function for BO is defined as the mean squared error (MSE) between the surrogate displacement field and snapshot observation. The inverse estimation capability of the proposed approach is tested in three different isotropic media with different wave velocities. From the obtained results, we have observed that BOPINN can accurately estimate wave velocities with lower MSE, even in the presence of noisy conditions. The proposed algorithm shows robust predictions in limited iterations across different runs.
Being the most classical generative model for serial data, state-space models (SSM) are fundamental in AI and statistical machine learning. In SSM, any form of parameter learning or latent state inference typically involves the computation of complex latent-state posteriors. In this work, we build upon the variational sequential Monte Carlo (VSMC) method, which provides computationally efficient and accurate model parameter estimation and Bayesian latent-state inference by combining particle methods and variational inference. While standard VSMC operates in the offline mode, by re-processing repeatedly a given batch of data, we distribute the approximation of the gradient of the VSMC surrogate ELBO in time using stochastic approximation, allowing for online learning in the presence of streams of data. This results in an algorithm, online VSMC, that is capable of performing efficiently, entirely on-the-fly, both parameter estimation and particle proposal adaptation. In addition, we provide rigorous theoretical results describing the algorithm's convergence properties as the number of data tends to infinity as well as numerical illustrations of its excellent convergence properties and usefulness also in batch-processing settings.
One of the pivotal tasks in scientific machine learning is to represent underlying dynamical systems from time series data. Many methods for such dynamics learning explicitly require the derivatives of state data, which are not directly available and can be approximated conventionally by finite differences. However, the discrete approximations of time derivatives may result in poor estimations when state data are scarce and/or corrupted by noise, thus compromising the predictiveness of the learned dynamical models. To overcome this technical hurdle, we propose a new method that learns nonlinear dynamics through a Bayesian inference of characterizing model parameters. This method leverages a Gaussian process representation of states, and constructs a likelihood function using the correlation between state data and their derivatives, yet prevents explicit evaluations of time derivatives. Through a Bayesian scheme, a probabilistic estimate of the model parameters is given by the posterior distribution, and thus a quantification is facilitated for uncertainties from noisy state data and the learning process. Specifically, we will discuss the applicability of the proposed method to several typical scenarios for dynamical systems: identification and estimation with an affine parametrization, nonlinear parametric approximation without prior knowledge, and general parameter estimation for a given dynamical system.
We present a novel deep learning method for estimating time-dependent parameters in Markov processes through discrete sampling. Departing from conventional machine learning, our approach reframes parameter approximation as an optimization problem using the maximum likelihood approach. Experimental validation focuses on parameter estimation in multivariate regression and stochastic differential equations (SDEs). Theoretical results show that the real solution is close to SDE with parameters approximated using our neural network-derived under specific conditions. Our work contributes to SDE-based model parameter estimation, offering a versatile tool for diverse fields.
In this paper we introduce GP+, an open-source library for kernel-based learning via Gaussian processes (GPs) which are powerful statistical models that are completely characterized by their parametric covariance and mean functions. GP+ is built on PyTorch and provides a user-friendly and object-oriented tool for probabilistic learning and inference. As we demonstrate with a host of examples, GP+ has a few unique advantages over other GP modeling libraries. We achieve these advantages primarily by integrating nonlinear manifold learning techniques with GPs' covariance and mean functions. As part of introducing GP+, in this paper we also make methodological contributions that (1) enable probabilistic data fusion and inverse parameter estimation, and (2) equip GPs with parsimonious parametric mean functions which span mixed feature spaces that have both categorical and quantitative variables. We demonstrate the impact of these contributions in the context of Bayesian optimization, multi-fidelity modeling, sensitivity analysis, and calibration of computer models.
Likelihood-free inference is quickly emerging as a powerful tool to perform fast/effective parameter estimation. We demonstrate a technique of optimizing likelihood-free inference to make it even faster by marginalizing symmetries in a physical problem. In this approach, physical symmetries, for example, time-translation are learned using joint-embedding via self-supervised learning with symmetry data augmentations. Subsequently, parameter inference is performed using a normalizing flow where the embedding network is used to summarize the data before conditioning the parameters. We present this approach on two simple physical problems and we show faster convergence in a smaller number of parameters compared to a normalizing flow that does not use a pre-trained symmetry-informed representation.
Recently, a novel system identification method based on invariant subspace theory is introduced, aiming to address the identification problem of continuous-time (CT) linear time-invariant (LTI) systems by combining time-domain and frequency-domain methods. Subsequently, the combined Invariant-Subspace and Subspace Identification Method (cISSIM) is introduced, enabling direct estimation of CT LTI systems in state-space forms. It produces consistent estimation that is robust in an error-in-variable and slow-sampling conditions, while no pre-filtering operation of the input-output signals is needed. This paper presents the discrete-cISSIM, which extends cISSIM to discrete-time (DT) systems and offers the following improvements: 1) the capability to utilize arbitrary discrete periodic excitations while cISSIM uses multi-sine signals; 2) a faster estimation with reduced computational complexity is proposed; 3) the covariance estimation problem can be addressed concurrently with the system parameter estimation. An implementation of discrete-cISSIM by MATLAB has also been provided.
Differentiable particle filters are an emerging class of sequential Bayesian inference techniques that use neural networks to construct components in state space models. Existing approaches are mostly based on offline supervised training strategies. This leads to the delay of the model deployment and the obtained filters are susceptible to distribution shift of test-time data. In this paper, we propose an online learning framework for differentiable particle filters so that model parameters can be updated as data arrive. The technical constraint is that there is no known ground truth state information in the online inference setting. We address this by adopting an unsupervised loss to construct the online model updating procedure, which involves a sequence of filtering operations for online maximum likelihood-based parameter estimation. We empirically evaluate the effectiveness of the proposed method, and compare it with supervised learning methods in simulation settings including a multivariate linear Gaussian state-space model and a simulated object tracking experiment.
A mixture of multivariate Poisson-log normal factor analyzers is introduced by imposing constraints on the covariance matrix, which resulted in flexible models for clustering purposes. In particular, a class of eight parsimonious mixture models based on the mixtures of factor analyzers model are introduced. Variational Gaussian approximation is used for parameter estimation, and information criteria are used for model selection. The proposed models are explored in the context of clustering discrete data arising from RNA sequencing studies. Using real and simulated data, the models are shown to give favourable clustering performance. The GitHub R package for this work is available at https://github.com/anjalisilva/mixMPLNFA and is released under the open-source MIT license.
In this paper, we show how convolutional neural networks (CNN) can be used in regression and classification learning problems of noisy and non-noisy functional data. The main idea is to transform the functional data into a 28 by 28 image. We use a specific but typical architecture of a convolutional neural network to perform all the regression exercises of parameter estimation and functional form classification. First, we use some functional case studies of functional data with and without random noise to showcase the strength of the new method. In particular, we use it to estimate exponential growth and decay rates, the bandwidths of sine and cosine functions, and the magnitudes and widths of curve peaks. We also use it to classify the monotonicity and curvatures of functional data, algebraic versus exponential growth, and the number of peaks of functional data. Second, we apply the same convolutional neural networks to Lyapunov exponent estimation in noisy and non-noisy chaotic data, in estimating rates of disease transmission from epidemic curves, and in detecting the similarity of drug dissolution profiles. Finally, we apply the method to real-life data to detect Parkinson's disease patients in a classification problem. The method, although simple, shows high accuracy and is promising for future use in engineering and medical applications.
Discovering mathematical equations that govern physical and biological systems from observed data is a fundamental challenge in scientific research. We present a new physics-informed framework for parameter estimation and missing physics identification (gray-box) in the field of Systems Biology. The proposed framework -- named AI-Aristotle -- combines eXtreme Theory of Functional Connections (X-TFC) domain-decomposition and Physics-Informed Neural Networks (PINNs) with symbolic regression (SR) techniques for parameter discovery and gray-box identification. We test the accuracy, speed, flexibility and robustness of AI-Aristotle based on two benchmark problems in Systems Biology: a pharmacokinetics drug absorption model, and an ultradian endocrine model for glucose-insulin interactions. We compare the two machine learning methods (X-TFC and PINNs), and moreover, we employ two different symbolic regression techniques to cross-verify our results. While the current work focuses on the performance of AI-Aristotle based on synthetic data, it can equally handle noisy experimental data and can even be used for black-box identification in just a few minutes on a laptop. More broadly, our work provides insights into the accuracy, cost, scalability, and robustness of integrating neural networks with symbolic regressors, offering a comprehensive guide for researchers tackling gray-box identification challenges in complex dynamical systems in biomedicine and beyond.
This study presents a novel generative modeling approach to rainfall-runoff modeling, focusing on the synthesis of realistic daily catchment runoff time series in response to catchment-averaged climate forcing. Unlike traditional process-based lumped hydrologic models that depend on predefined sets of variables describing catchment physical properties, our approach uses a small number of latent variables to characterize runoff generation processes. These latent variables encapsulate the intrinsic properties of a catchment and can be inferred from catchment climate forcing and discharge data. By sampling from the latent variable space, the model generates runoff time series that closely resemble real-world observations. In this study, we trained the generative models using neural networks on data from over 3,000 global catchments and achieved prediction accuracies comparable to current deep learning models and various conventional lumped models, both within the catchments from the training set and from other regions worldwide. This suggests that the runoff generation process of catchments can be effectively captured by a low-dimensional latent representation. Yet, challenges such as equifinality and optimal determination of latent variables remain. Future research should focus on refining parameter estimation methods and exploring the physical meaning of these latent dimensions to improve model applicability and robustness. This generative approach offers a promising alternative for hydrological modeling that requires minimal assumptions about the physical processes of the catchment.
Energy forecasting is an essential task in power system operations. Operators usually issue forecasts and leverage them to schedule energy dispatch ahead of time. However, forecast models are typically developed in a way that overlooks the operational value of the forecasts. To bridge the gap, we design a value-oriented point forecasting approach for sequential energy dispatch problems with renewable energy sources. At the training phase, we align the loss function with the overall operation cost function, thereby achieving reduced operation costs. The forecast model parameter estimation is formulated as a bilevel program. Under mild assumptions, we convert the upper-level objective into an equivalent form using the dual solutions obtained from the lower-level operation problems. Additionally, a novel iterative solution strategy is proposed for the newly formulated bilevel program. Under such an iterative scheme, we show that the upper-level objective is locally linear regarding the forecast model output, and can act as the loss function. Numerical experiments demonstrate that, compared to commonly used statistical quality-oriented point forecasting methods, forecasts obtained by the proposed approach result in lower operation costs. Meanwhile, the proposed approach is more computationally efficient than traditional two-stage stochastic programs.
The analysis of Synthetic Aperture Radar (SAR) imagery is an important step in remote sensing applications, and it is a challenging problem due to its inherent speckle noise. One typical solution is to model the data using the $G_I^0$ distribution and extract its roughness information, which in turn can be used in posterior imaging tasks, such as segmentation, classification and interpretation. This leads to the need of quick and reliable estimation of the roughness parameter from SAR data, especially with high resolution images. Unfortunately, traditional parameter estimation procedures are slow and prone to estimation failures. In this work, we proposed a neural network-based estimation framework that first learns how to predict underlying parameters of $G_I^0$ samples and then can be used to estimate the roughness of unseen data. We show that this approach leads to an estimator that is quicker, yields less estimation error and is less prone to failures than the traditional estimation procedures for this problem, even when we use a simple network. More importantly, we show that this same methodology can be generalized to handle image inputs and, even if trained on purely synthetic data for a few seconds, is able to perform real time pixel-wise roughness estimation for high resolution real SAR imagery.
The image source method (ISM) is often used to simulate room acoustics due to its ease of use and computational efficiency. The standard ISM is limited to simulations of room impulse responses between point sources and omnidirectional receivers. In this work, the ISM is extended using spherical harmonic directivity coefficients to include acoustic diffraction effects due to source and receiver transducers mounted on physical devices, which are typically encountered in practical situations. The proposed method is verified using finite element simulations of various loudspeaker and microphone configurations in a rectangular room. It is shown that the accuracy of the proposed method is related to the sizes, shapes, number, and positions of the devices inside a room. A simplified version of the proposed method, which can significantly reduce computational effort, is also presented. The proposed method and its simplified version can simulate room transfer functions more accurately than currently available image source methods and can aid the development and evaluation of speech and acoustic signal processing algorithms, including speech enhancement, acoustic scene analysis, and acoustic parameter estimation.