Pouria Behnoudfar

Quanling Deng, Pouria Behnoudfar, Victor M. Calo

The generalized-$α$ method encompasses a wide range of time integrators. The method possesses high-frequency dissipation while minimizing unwanted low-frequency dissipation and the numerical dissipation can be controlled by the user. The method is unconditionally stable and is of second-order accuracy in time. We extend the second-order generalized-$α$ method to third-order in time while the numerical dissipation can be controlled in a similar fashion. We establish that the third-order method is unconditionally stable. We discuss a possible path to the generalization to higher order schemes. All these high-order schemes can be easily implemented into programs that already contain the second-order generalized-$α$ method.

Pouria Behnoudfar, Victor M. Calo, Quanling Deng et al.

We present a variationally separable splitting technique for the generalized-$α$ method for solving parabolic partial differential equations. We develop a technique for a tensor-product mesh which results in a solver with a linear cost with respect to the total number of degrees of freedom in the system for multi-dimensional problems. We consider finite elements and isogeometric analysis for the spatial discretization. The overall method maintains user-controlled high-frequency dissipation while minimizing unwanted low-frequency dissipation. The method has second-order accuracy in time and optimal rates ($h^{p+1}$ in $L^2$ norm and $h^p$ in $L^2$ norm of $\nabla u$) in space. We present the spectrum analysis on the amplification matrix to establish that the method is unconditionally stable. Various numerical examples illustrate the performance of the overall methodology and show the optimal approximation accuracy.

Luis Espath, Pouria Behnoudfar, Raul Tempone

In this work, we further develop the Physics-informed Spectral Learning (PiSL) by Espath et al. \cite{Esp21} based on a discrete $L^2$ projection to solve the discrete Hodge--Helmholtz decomposition from sparse data. Within this physics-informed statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients by solving a sequence of minimization problems where the set of basis functions is augmented greedily at each optimization problem. Moreover, our PiSL computational framework enjoys spectral (exponential) convergence. We regularize the minimization problems with the seminorm of the fractional Sobolev space in a Tikhonov fashion. In the Fourier setting, the divergence- and curl-free constraints become a finite set of linear algebraic equations. The proposed computational framework combines supervised and unsupervised learning techniques in that we use data concomitantly with the projection onto divergence- and curl-free spaces. We assess the capabilities of our method in various numerical examples including the `Storm of the Century' with satellite data from 1993.

Pouria Behnoudfar, Deekshith Naidu Ponnana, Noah J. Schmelzer et al.

We present a probabilistic modeling framework for incorporating small-scale spatial heterogeneity into macroscopic descriptions of material behavior for polycrystalline metallic materials. Spatially heterogeneous material state fields are represented using probability density functions (PDFs), providing a principled statistical description of microstructural variability and state evolution across different computational polycrystalline realizations. The framework is built on the inverse identification of a probabilistic transport model, formulated as a Liouville equation with an unknown drift term. To enable accurate, stable, and interpretable inference of this drift field in high-dimensional, transport-dominated settings, we develop a Separable Probability Learning Technique via Physics-Informed Neural Networks (SPLIT-PINN). This method incorporates a marginal-correction drift decomposition, orthogonality constraints, and residual-based adaptive training to enhance well-posedness, numerical stability, and physical consistency without imposing restrictive parametric assumptions. Using SPLIT-PINN, the drift field governing the temporal evolution of joint state PDFs is inferred directly from data. After benchmark validation, the framework is applied to physical computational datasets describing the evolution of polycrystalline microstructural states, including von Mises stress, dislocation density, and equivalent plastic strain rate. The learned Liouville model, trained on a single dataset, is subsequently used in forward predictions of the temporal evolution of joint and marginal PDFs for multiple unseen polycrystal realizations. Quantitative comparisons with reference PDFs demonstrate that the proposed framework yields accurate and robust probabilistic predictions and generalizes effectively across datasets.

Md Rakibul Hasan, Pouria Behnoudfar, Dan MacKinlay et al.

Machine Learning, particularly Generative Adversarial Networks (GANs), has revolutionised Super-Resolution (SR). However, generated images often lack physical meaningfulness, which is essential for scientific applications. Our approach, PC-SRGAN, enhances image resolution while ensuring physical consistency for interpretable simulations. PC-SRGAN significantly improves both the Peak Signal-to-Noise Ratio and the Structural Similarity Index Measure compared to conventional SR methods, even with limited training data (e.g., only 13% of training data is required to achieve performance similar to SRGAN). Beyond SR, PC-SRGAN augments physically meaningful machine learning, incorporating numerically justified time integrators and advanced quality metrics. These advancements promise reliable and causal machine-learning models in scientific domains. A significant advantage of PC-SRGAN over conventional SR techniques is its physical consistency, which makes it a viable surrogate model for time-dependent problems. PC-SRGAN advances scientific machine learning by improving accuracy and efficiency, enhancing process understanding, and broadening applications to scientific research. We publicly release the complete source code of PC-SRGAN and all experiments at https://github.com/hasan-rakibul/PC-SRGAN.

Pouria Behnoudfar, Nan Chen

Machine learning has become a powerful tool for enhancing data assimilation. While supervised learning remains the standard method, reinforcement learning (RL) offers unique advantages through its sequential decision-making framework, which naturally fits the iterative nature of data assimilation by dynamically balancing model forecasts with observations. We develop RL-DAUNCE, a new RL-based method that enhances data assimilation with physical constraints through three key aspects. First, RL-DAUNCE inherits the computational efficiency of machine learning while it uniquely structures its agents to mirror ensemble members in conventional data assimilation methods. Second, RL-DAUNCE emphasizes uncertainty quantification by advancing multiple ensemble members, moving beyond simple mean-state optimization. Third, RL-DAUNCE's ensemble-as-agents design facilitates the enforcement of physical constraints during the assimilation process, which is crucial to improving the state estimation and subsequent forecasting. A primal-dual optimization strategy is developed to enforce constraints, which dynamically penalizes the reward function to ensure constraint satisfaction throughout the learning process. Also, state variable bounds are respected by constraining the RL action space. Together, these features ensure physical consistency without sacrificing efficiency. RL-DAUNCE is applied to the Madden-Julian Oscillation, an intermittent atmospheric phenomenon characterized by strongly non-Gaussian features and multiple physical constraints. RL-DAUNCE outperforms the standard ensemble Kalman filter (EnKF), which fails catastrophically due to the violation of physical constraints. Notably, RL-DAUNCE matches the performance of constrained EnKF, particularly in recovering intermittent signals, capturing extreme events, and quantifying uncertainties, while requiring substantially less computational effort.

Pouria Behnoudfar, Charlotte Moser, Marc Bocquet et al.

Computer models are indispensable tools for understanding the Earth system. While high-resolution operational models have achieved many successes, they exhibit persistent biases, particularly in simulating extreme events and statistical distributions. In contrast, coarse-grained idealized models isolate fundamental processes and can be precisely calibrated to excel in characterizing specific dynamical and statistical features. However, different models remain siloed by disciplinary boundaries. By leveraging the complementary strengths of models of varying complexity, we develop an explainable AI framework for Earth system emulators. It bridges the model hierarchy through a reconfigured latent data assimilation technique, uniquely suited to exploit the sparse output from the idealized models. The resulting bridging model inherits the high resolution and comprehensive variables of operational models while achieving global accuracy enhancements through targeted improvements from idealized models. Crucially, the mechanism of AI provides a clear rationale for these advancements, moving beyond black-box correction to physically insightful understanding in a computationally efficient framework that enables effective physics-assisted digital twins and uncertainty quantification. We demonstrate its power by significantly correcting biases in CMIP6 simulations of El Niño spatiotemporal patterns, leveraging statistically accurate idealized models. This work also highlights the importance of pushing idealized model development and advancing communication between modeling communities.