Jérôme Darbon

Paula Chen, Tingwei Meng, Zongren Zou et al.

We address two major challenges in scientific machine learning (SciML): interpretability and computational efficiency. We increase the interpretability of certain learning processes by establishing a new theoretical connection between optimization problems arising from SciML and a generalized Hopf formula, which represents the viscosity solution to a Hamilton-Jacobi partial differential equation (HJ PDE) with time-dependent Hamiltonian. Namely, we show that when we solve certain regularized learning problems with integral-type losses, we actually solve an optimal control problem and its associated HJ PDE with time-dependent Hamiltonian. This connection allows us to reinterpret incremental updates to learned models as the evolution of an associated HJ PDE and optimal control problem in time, where all of the previous information is intrinsically encoded in the solution to the HJ PDE. As a result, existing HJ PDE solvers and optimal control algorithms can be reused to design new efficient training approaches for SciML that naturally coincide with the continual learning framework, while avoiding catastrophic forgetting. As a first exploration of this connection, we consider the special case of linear regression and leverage our connection to develop a new Riccati-based methodology for solving these learning problems that is amenable to continual learning applications. We also provide some corresponding numerical examples that demonstrate the potential computational and memory advantages our Riccati-based approach can provide.

Paula Chen, Tingwei Meng, Zongren Zou et al.

Hamilton-Jacobi partial differential equations (HJ PDEs) have deep connections with a wide range of fields, including optimal control, differential games, and imaging sciences. By considering the time variable to be a higher dimensional quantity, HJ PDEs can be extended to the multi-time case. In this paper, we establish a novel theoretical connection between specific optimization problems arising in machine learning and the multi-time Hopf formula, which corresponds to a representation of the solution to certain multi-time HJ PDEs. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multi-time HJ PDE and, by extension, its corresponding optimal control problem. As a first exploration of this connection, we develop the relation between the regularized linear regression problem and the Linear Quadratic Regulator (LQR). We then leverage our theoretical connection to adapt standard LQR solvers (namely, those based on the Riccati ordinary differential equations) to design new training approaches for machine learning. Finally, we provide some numerical examples that demonstrate the versatility and possible computational advantages of our Riccati-based approach in the context of continual learning, post-training calibration, transfer learning, and sparse dynamics identification.

Tingwei Meng, Zongren Zou, Jérôme Darbon et al.

The interplay between stochastic processes and optimal control has been extensively explored in the literature. With the recent surge in the use of diffusion models, stochastic processes have increasingly been applied to sample generation. This paper builds on the log transform, known as the Cole-Hopf transform in Brownian motion contexts, and extends it within a more abstract framework that includes a linear operator. Within this framework, we found that the well-known relationship between the Cole-Hopf transform and optimal transport is a particular instance where the linear operator acts as the infinitesimal generator of a stochastic process. We also introduce a novel scenario where the linear operator is the adjoint of the generator, linking to Bayesian inference under specific initial and terminal conditions. Leveraging this theoretical foundation, we develop a new algorithm, named the HJ-sampler, for Bayesian inference for the inverse problem of a stochastic differential equation with given terminal observations. The HJ-sampler involves two stages: (1) solving the viscous Hamilton-Jacobi partial differential equations, and (2) sampling from the associated stochastic optimal control problem. Our proposed algorithm naturally allows for flexibility in selecting the numerical solver for viscous HJ PDEs. We introduce two variants of the solver: the Riccati-HJ-sampler, based on the Riccati method, and the SGM-HJ-sampler, which utilizes diffusion models. We demonstrate the effectiveness and flexibility of the proposed methods by applying them to solve Bayesian inverse problems involving various stochastic processes and prior distributions, including applications that address model misspecifications and quantifying model uncertainty.

Zhen Zhang, Shanqing Liu, Alessandro Alla et al.

We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations. We first recall the classical direct and indirect viewpoints for optimal control of PDEs, and then present two PINN formulations: a direct formulation based on minimizing the objective under the state constraint, and an indirect formulation based on the first-order optimality system. For a class of semilinear parabolic equations, we derive the state equation, the adjoint equation, and the stationarity condition in a form consistent with continuous-time Pontryagin-type optimality conditions. We then specialize the framework to an Allen-Cahn control problem and compare three numerical approaches: (i) a discretize-then-optimize adjoint method, (ii) a direct PINN, and (iii) an indirect PINN. Numerical results show that the PINN parameterization has an implicit regularizing effect, in the sense that it tends to produce smoother control profiles. They also indicate that the indirect PINN more faithfully preserves the PDE contraint and optimality structure and yields a more accurate neural approximation than the direct PINN.

Youngkyu Lee, Shanqing Liu, Jerome Darbon et al.

We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterations caused by unbalanced nonlinearities, we introduce a fixed-point neural operator (FPNO) that learns the direct mapping from the current iterate to the solution by emulating fixed-point iterations. Unlike traditional line-search or trust-region algorithms, the proposed FPNO adaptively employs negative step sizes to effectively mitigate the effects of unbalanced nonlinearities. Through numerical experiments we demonstrate the computational efficiency and robustness of the proposed NP-Newton method across multiple real-world applications, especially for very strong nonlinearities.

Elham Kiyani, Khemraj Shukla, Jorge F. Urbán et al.

Physics-Informed Neural Networks (PINNs) have revolutionized the computation of PDE solutions by integrating partial differential equations (PDEs) into the neural network's training process as soft constraints, becoming an important component of the scientific machine learning (SciML) ecosystem. More recently, physics-informed Kolmogorv-Arnold networks (PIKANs) have also shown to be effective and comparable in accuracy with PINNs. In their current implementation, both PINNs and PIKANs are mainly optimized using first-order methods like Adam, as well as quasi-Newton methods such as BFGS and its low-memory variant, L-BFGS. However, these optimizers often struggle with highly non-linear and non-convex loss landscapes, leading to challenges such as slow convergence, local minima entrapment, and (non)degenerate saddle points. In this study, we investigate the performance of Self-Scaled BFGS (SSBFGS), Self-Scaled Broyden (SSBroyden) methods and other advanced quasi-Newton schemes, including BFGS and L-BFGS with different line search strategies. These methods dynamically rescale updates based on historical gradient information, thus enhancing training efficiency and accuracy. We systematically compare these optimizers using both PINNs and PIKANs on key challenging PDEs, including the Burgers, Allen-Cahn, Kuramoto-Sivashinsky, Ginzburg-Landau, and Stokes equations. Additionally, we evaluate the performance of SSBFGS and SSBroyden for Deep Operator Network (DeepONet) architectures, demonstrating their effectiveness for data-driven operator learning. Our findings provide state-of-the-art results with orders-of-magnitude accuracy improvements without the use of adaptive weights or any other enhancements typically employed in PINNs.

Jaemin Oh, Youngkyu Lee, Jerome Darbon et al.

Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction. We show that this premise is unreliable. An operator trained to the relative \(L^2\) error \(O(10^{-3})\) can still produce an initial state in which the discrete Jacobian is indefinite, because the mean-squared training controls error on average while leaving localized pointwise violations of the underlying physics. For a nearly incompressible hyperelasticity problem, we trace this to the predicted volume change: the operator disperses \(\mathrm{det} F\) well away from one, and the resulting Jacobian acquires negative eigenvalues even when the predicted field is visually indistinguishable from the reference. At a small scale, this is a nuisance; at a multi-million degree-of-freedom scale, it is disqualifying, since the conjugate gradient and other Krylov solvers needed for memory-feasible Newton steps assume a definite spectrum. We then show that a short, label-free fine-tuning phase -- penalizing the operator against the discrete energy, with no additional solution data -- shifts the Jacobian spectrum back to positive definite. Combined with an inexact outer loop, this gives a warm-started Newton method that converges across the full loading range where the unregularized operator fails, reaching up to 5.4\(\times\) wall-clock speedup over incremental continuation on a 3D problem with 6.4 million degrees of freedom.

Zongren Zou, Tingwei Meng, Paula Chen et al.

Uncertainty quantification (UQ) in scientific machine learning (SciML) combines the powerful predictive power of SciML with methods for quantifying the reliability of the learned models. However, two major challenges remain: limited interpretability and expensive training procedures. We provide a new interpretation for UQ problems by establishing a new theoretical connection between some Bayesian inference problems arising in SciML and viscous Hamilton-Jacobi partial differential equations (HJ PDEs). Namely, we show that the posterior mean and covariance can be recovered from the spatial gradient and Hessian of the solution to a viscous HJ PDE. As a first exploration of this connection, we specialize to Bayesian inference problems with linear models, Gaussian likelihoods, and Gaussian priors. In this case, the associated viscous HJ PDEs can be solved using Riccati ODEs, and we develop a new Riccati-based methodology that provides computational advantages when continuously updating the model predictions. Specifically, our Riccati-based approach can efficiently add or remove data points to the training set invariant to the order of the data and continuously tune hyperparameters. Moreover, neither update requires retraining on or access to previously incorporated data. We provide several examples from SciML involving noisy data and \textit{epistemic uncertainty} to illustrate the potential advantages of our approach. In particular, this approach's amenability to data streaming applications demonstrates its potential for real-time inferences, which, in turn, allows for applications in which the predicted uncertainty is used to dynamically alter the learning process.

Qianying Cao, Shanqing Liu, Alan John Varghese et al.

Time series forecasting plays a pivotal role in a wide range of applications, including weather prediction, healthcare, structural health monitoring, predictive maintenance, energy systems, and financial markets. While models such as LSTM, GRU, Transformers, and State-Space Models (SSMs) have become standard tools in this domain, selecting the optimal architecture remains a challenge. Performance comparisons often depend on evaluation metrics and the datasets under analysis, making the choice of a universally optimal model controversial. In this work, we introduce a flexible automated framework for time series forecasting that systematically designs and evaluates diverse network architectures by integrating LSTM, GRU, multi-head Attention, and SSM blocks. Using a multi-objective optimization approach, our framework determines the number, sequence, and combination of blocks to align with specific requirements and evaluation objectives. From the resulting Pareto-optimal architectures, the best model for a given context is selected via a user-defined preference function. We validate our framework across four distinct real-world applications. Results show that a single-layer GRU or LSTM is usually optimal when minimizing training time alone. However, when maximizing accuracy or balancing multiple objectives, the best architectures are often composite designs incorporating multiple block types in specific configurations. By employing a weighted preference function, users can resolve trade-offs between objectives, revealing novel, context-specific optimal architectures. Our findings underscore that no single neural architecture is universally optimal for time series forecasting. Instead, the best-performing model emerges as a data-driven composite architecture tailored to user-defined criteria and evaluation objectives.

Gabriel P. Langlois, Jatan Buch, Jérôme Darbon

Maximum entropy (Maxent) models are a class of statistical models that use the maximum entropy principle to estimate probability distributions from data. Due to the size of modern data sets, Maxent models need efficient optimization algorithms to scale well for big data applications. State-of-the-art algorithms for Maxent models, however, were not originally designed to handle big data sets; these algorithms either rely on technical devices that may yield unreliable numerical results, scale poorly, or require smoothness assumptions that many practical Maxent models lack. In this paper, we present novel optimization algorithms that overcome the shortcomings of state-of-the-art algorithms for training large-scale, non-smooth Maxent models. Our proposed first-order algorithms leverage the Kullback-Leibler divergence to train large-scale and non-smooth Maxent models efficiently. For Maxent models with discrete probability distribution of $n$ elements built from samples, each containing $m$ features, the stepsize parameters estimation and iterations in our algorithms scale on the order of $O(mn)$ operations and can be trivially parallelized. Moreover, the strong $\ell_{1}$ convexity of the Kullback--Leibler divergence allows for larger stepsize parameters, thereby speeding up the convergence rate of our algorithms. To illustrate the efficiency of our novel algorithms, we consider the problem of estimating probabilities of fire occurrences as a function of ecological features in the Western US MTBS-Interagency wildfire data set. Our numerical results show that our algorithms outperform the state of the arts by one order of magnitude and yield results that agree with physical models of wildfire occurrence and previous statistical analyses of wildfire drivers.

Juan Diego Toscano, Daniel T. Chen, Vivek Oommen et al.

Residual-based adaptive strategies are widely used in scientific machine learning but remain largely heuristic. We introduce a unifying variational framework that formalizes these methods by integrating convex transformations of the residual. Different transformations correspond to distinct objective functionals: exponential weights target the minimization of uniform error, while linear weights recover the minimization of quadratic error. Within this perspective, adaptive weighting is equivalent to selecting sampling distributions that optimize the primal objective, thereby linking discretization choices directly to error metrics. This principled approach yields three benefits: (1) it enables systematic design of adaptive schemes across norms, (2) reduces discretization error through variance reduction of the loss estimator, and (3) enhances learning dynamics by improving the gradient signal-to-noise ratio. Extending the framework to operator learning, we demonstrate substantial performance gains across optimizers and architectures. Our results provide a theoretical justification of residual-based adaptivity and establish a foundation for principled discretization and training strategies.

Gabriel P. Langlois, Jérôme Darbon

We prove in this work that the well-known lasso problem can be solved exactly without homotopy using novel differential inclusions techniques. Specifically, we show that a selection principle from the theory of differential inclusions transforms the dual lasso problem into the problem of calculating the trajectory of a projected dynamical system that we prove is integrable. Our analysis yields an exact algorithm for the lasso problem, numerically up to machine precision, that is amenable to computing regularization paths and is very fast. Moreover, we show the continuation of solutions to the integrable projected dynamical system in terms of the hyperparameter naturally yields a rigorous homotopy algorithm. Numerical experiments confirm that our algorithm outperforms the state-of-the-art algorithms in both efficiency and accuracy. Beyond this work, we expect our results and analysis can be adapted to compute exact or approximate solutions to a broader class of polyhedral-constrained optimization problems.

Tingwei Meng, Zhen Zhang, Jérôme Darbon et al.

Solving high-dimensional optimal control problems in real-time is an important but challenging problem, with applications to multi-agent path planning problems, which have drawn increased attention given the growing popularity of drones in recent years. In this paper, we propose a novel neural network method called SympOCnet that applies the Symplectic network to solve high-dimensional optimal control problems with state constraints. We present several numerical results on path planning problems in two-dimensional and three-dimensional spaces. Specifically, we demonstrate that our SympOCnet can solve a problem with more than 500 dimensions in 1.5 hours on a single GPU, which shows the effectiveness and efficiency of SympOCnet. The proposed method is scalable and has the potential to solve truly high-dimensional path planning problems in real-time.

Jérôme Darbon, Gabriel P. Langlois

Logistic regression is a widely used statistical model to describe the relationship between a binary response variable and predictor variables in data sets. It is often used in machine learning to identify important predictor variables. This task, variable selection, typically amounts to fitting a logistic regression model regularized by a convex combination of $\ell_1$ and $\ell_{2}^{2}$ penalties. Since modern big data sets can contain hundreds of thousands to billions of predictor variables, variable selection methods depend on efficient and robust optimization algorithms to perform well. State-of-the-art algorithms for variable selection, however, were not traditionally designed to handle big data sets; they either scale poorly in size or are prone to produce unreliable numerical results. It therefore remains challenging to perform variable selection on big data sets without access to adequate and costly computational resources. In this paper, we propose a nonlinear primal-dual algorithm that addresses these shortcomings. Specifically, we propose an iterative algorithm that provably computes a solution to a logistic regression problem regularized by an elastic net penalty in $O(T(m,n)\log(1/ε))$ operations, where $ε\in (0,1)$ denotes the tolerance and $T(m,n)$ denotes the number of arithmetic operations required to perform matrix-vector multiplication on a data set with $m$ samples each comprising $n$ features. This result improves on the known complexity bound of $O(\min(m^2n,mn^2)\log(1/ε))$ for first-order optimization methods such as the classic primal-dual hybrid gradient or forward-backward splitting methods.

Jérôme Darbon, Gabriel P. Langlois

The linear primal-dual hybrid gradient (PDHG) method is a first-order method that splits convex optimization problems with saddle-point structure into smaller subproblems. Unlike those obtained in most splitting methods, these subproblems can generally be solved efficiently because they involve simple operations such as matrix-vector multiplications or proximal mappings that are fast to evaluate numerically. This advantage comes at the price that the linear PDHG method requires precise stepsize parameters for the problem at hand to achieve an optimal convergence rate. Unfortunately, these stepsize parameters are often prohibitively expensive to compute for large-scale optimization problems, such as those in machine learning. This issue makes the otherwise simple linear PDHG method unsuitable for such problems, and it is also shared by most first-order optimization methods as well. To address this issue, we introduce accelerated nonlinear PDHG methods that achieve an optimal convergence rate with stepsize parameters that are simple and efficient to compute. We prove rigorous convergence results, including results for strongly convex or smooth problems posed on infinite-dimensional reflexive Banach spaces. We illustrate the efficiency of our methods on $\ell_{1}$-constrained logistic regression and entropy-regularized matrix games. Our numerical experiments show that the nonlinear PDHG methods are considerably faster than competing methods.

Jérôme Darbon, Tingwei Meng, Elena Resmerita

We consider image denoising problems formulated as variational problems. It is known that Hamilton-Jacobi PDEs govern the solution of such optimization problems when the noise model is additive. In this work, we address certain non-additive noise models and show that they are also related to Hamilton-Jacobi PDEs. These findings allow us to establish new connections between additive and non-additive noise imaging models. Specifically, we study how the solutions to these optimization problems depend on the parameters and the observed images. We show that the optimal values are ruled by some Hamilton-Jacobi PDEs, while the optimizers are characterized by the spatial gradient of the solution to the Hamilton-Jacobi PDEs. Moreover, we use these relations to investigate the asymptotic behavior of the variational model as the parameter goes to infinity, that is, when the influence of the noise vanishes. With these connections, some non-convex models for non-additive noise can be solved by applying convex optimization algorithms to the equivalent convex models for additive noise. Several numerical results are provided for denoising problems with Poisson noise or multiplicative noise.

Jérôme Darbon, Peter M. Dower, Tingwei Meng

Solving high dimensional optimal control problems and corresponding Hamilton-Jacobi PDEs are important but challenging problems in control engineering. In this paper, we propose two abstract neural network architectures which are respectively used to compute the value function and the optimal control for certain class of high dimensional optimal control problems. We provide the mathematical analysis for the two abstract architectures. We also show several numerical results computed using the deep neural network implementations of these abstract architectures. A preliminary implementation of our proposed neural network architecture on FPGAs shows promising speed up compared to CPUs. This work paves the way to leverage efficient dedicated hardware designed for neural networks to solve high dimensional optimal control problems and Hamilton-Jacobi PDEs.

Jérôme Darbon, Gabriel P. Langlois, Tingwei Meng

Many imaging problems can be formulated as inverse problems expressed as finite-dimensional optimization problems. These optimization problems generally consist of minimizing the sum of a data fidelity and regularization terms. In [23,26], connections between these optimization problems and (multi-time) Hamilton--Jacobi partial differential equations have been proposed under the convexity assumptions of both the data fidelity and regularization terms. In particular, under these convexity assumptions, some representation formulas for a minimizer can be obtained. From a Bayesian perspective, such a minimizer can be seen as a maximum a posteriori estimator. In this chapter, we consider a certain class of non-convex regularizations and show that similar representation formulas for the minimizer can also be obtained. This is achieved by leveraging min-plus algebra techniques that have been originally developed for solving certain Hamilton--Jacobi partial differential equations arising in optimal control. Note that connections between viscous Hamilton--Jacobi partial differential equations and Bayesian posterior mean estimators with Gaussian data fidelity terms and log-concave priors have been highlighted in [25]. We also present similar results for certain Bayesian posterior mean estimators with Gaussian data fidelity and certain non-log-concave priors using an analogue of min-plus algebra techniques.

Yeonjong Shin, Jérôme Darbon, George Em Karniadakis

We propose a novel Caputo fractional derivative-based optimization algorithm. Upon defining the Caputo fractional gradient with respect to the Cartesian coordinate, we present a generic Caputo fractional gradient descent (CFGD) method. We prove that the CFGD yields the steepest descent direction of a locally smoothed objective function. The generic CFGD requires three parameters to be specified, and a choice of the parameters yields a version of CFGD. We propose three versions -- non-adaptive, adaptive terminal and adaptive order. By focusing on quadratic objective functions, we provide a convergence analysis. We prove that the non-adaptive CFGD converges to a Tikhonov regularized solution. For the two adaptive versions, we derive error bounds, which show convergence to integer-order stationary point under some conditions. We derive an explicit formula of CFGD for quadratic functions. We computationally found that the adaptive terminal (AT) CFGD mitigates the dependence on the condition number in the rate of convergence and results in significant acceleration over gradient descent (GD). For non-quadratic functions, we develop an efficient implementation of CFGD using the Gauss-Jacobi quadrature, whose computational cost is approximately proportional to the number of the quadrature points and the cost of GD. Our numerical examples show that AT-CFGD results in acceleration over GD, even when a small number of the Gauss-Jacobi quadrature points (including a single point) is used.