Saviz Mowlavi

Xiangyuan Zhang, Saviz Mowlavi, Mouhacine Benosman et al.

We introduce the receding-horizon policy gradient (RHPG) algorithm, the first PG algorithm with provable global convergence in learning the optimal linear estimator designs, i.e., the Kalman filter (KF). Notably, the RHPG algorithm does not require any prior knowledge of the system for initialization and does not require the target system to be open-loop stable. The key of RHPG is that we integrate vanilla PG (or any other policy search directions) into a dynamic programming outer loop, which iteratively decomposes the infinite-horizon KF problem that is constrained and non-convex in the policy parameter into a sequence of static estimation problems that are unconstrained and strongly-convex, thus enabling global convergence. We further provide fine-grained analyses of the optimization landscape under RHPG and detail the convergence and sample complexity guarantees of the algorithm. This work serves as an initial attempt to develop reinforcement learning algorithms specifically for control applications with performance guarantees by utilizing classic control theory in both algorithmic design and theoretical analyses. Lastly, we validate our theories by deploying the RHPG algorithm to learn the Kalman filter design of a large-scale convection-diffusion model. We open-source the code repository at \url{https://github.com/xiangyuan-zhang/LearningKF}.

Xiangyuan Zhang, Weichao Mao, Saviz Mowlavi et al.

We introduce controlgym, a library of thirty-six industrial control settings, and ten infinite-dimensional partial differential equation (PDE)-based control problems. Integrated within the OpenAI Gym/Gymnasium (Gym) framework, controlgym allows direct applications of standard reinforcement learning (RL) algorithms like stable-baselines3. Our control environments complement those in Gym with continuous, unbounded action and observation spaces, motivated by real-world control applications. Moreover, the PDE control environments uniquely allow the users to extend the state dimensionality of the system to infinity while preserving the intrinsic dynamics. This feature is crucial for evaluating the scalability of RL algorithms for control. This project serves the learning for dynamics & control (L4DC) community, aiming to explore key questions: the convergence of RL algorithms in learning control policies; the stability and robustness issues of learning-based controllers; and the scalability of RL algorithms to high- and potentially infinite-dimensional systems. We open-source the controlgym project at https://github.com/xiangyuan-zhang/controlgym.

Saviz Mowlavi, Mouhacine Benosman

In systems governed by nonlinear partial differential equations such as fluid flows, the design of state estimators such as Kalman filters relies on a reduced-order model (ROM) that projects the original high-dimensional dynamics onto a computationally tractable low-dimensional space. However, ROMs are prone to large errors, which negatively affects the performance of the estimator. Here, we introduce the reinforcement learning reduced-order estimator (RL-ROE), a ROM-based estimator in which the correction term that takes in the measurements is given by a nonlinear policy trained through reinforcement learning. The nonlinearity of the policy enables the RL-ROE to compensate efficiently for errors of the ROM, while still taking advantage of the imperfect knowledge of the dynamics. Using examples involving the Burgers and Navier-Stokes equations, we show that in the limit of very few sensors, the trained RL-ROE outperforms a Kalman filter designed using the same ROM. Moreover, it yields accurate high-dimensional state estimates for trajectories corresponding to various physical parameter values, without direct knowledge of the latter.

Saviz Mowlavi, Ken Kamrin

Most noninvasive imaging techniques utilize electromagnetic or acoustic waves originating from multiple locations and directions to identify hidden geometrical structures. Surprisingly, it is also possible to image hidden voids and inclusions buried within an object using a single static thermal or mechanical loading experiment by observing the response of the exposed surface of the body, but this problem is challenging to invert. Although physics-informed neural networks (PINNs) have shown promise as a simple-yet-powerful tool for problem inversion, they have not yet been applied to imaging problems with a priori unknown topology. Here, we introduce a topology optimization framework based on PINNs that identifies concealed geometries using exposed surface data from a single loading experiment, without prior knowledge of the number or types of shapes. We allow for arbitrary solution topology by representing the geometry using a material density field combined with a novel eikonal regularization technique. We validate our framework by detecting the number, locations, and shapes of hidden voids and inclusions in many example cases, in both 2D and 3D, and we demonstrate the method's robustness to noise and sparsity in the data. Our methodology opens a pathway for PINNs to solve geometry optimization problems in engineering.

Alexander Schperberg, Yusuke Tanaka, Saviz Mowlavi et al.

State estimation for legged robots is challenging due to their highly dynamic motion and limitations imposed by sensor accuracy. By integrating Kalman filtering, optimization, and learning-based modalities, we propose a hybrid solution that combines proprioception and exteroceptive information for estimating the state of the robot's trunk. Leveraging joint encoder and IMU measurements, our Kalman filter is enhanced through a single-rigid body model that incorporates ground reaction force control outputs from convex Model Predictive Control optimization. The estimation is further refined through Gated Recurrent Units, which also considers semantic insights and robot height from a Vision Transformer autoencoder applied on depth images. This framework not only furnishes accurate robot state estimates, including uncertainty evaluations, but can minimize the nonlinear errors that arise from sensor measurements and model simplifications through learning. The proposed methodology is evaluated in hardware using a quadruped robot on various terrains, yielding a 65% improvement on the Root Mean Squared Error compared to our VIO SLAM baseline. Code example: https://github.com/AlexS28/OptiState

Zhuoyuan Wang, Hanjiang Hu, Xiyu Deng et al.

Solving diverse partial differential equations (PDEs) is fundamental in science and engineering. Large language models (LLMs) have demonstrated strong capabilities in code generation, symbolic reasoning, and tool use, but reliably solving PDEs across heterogeneous settings remains challenging. Prior work on LLM-based code generation and transformer-based foundation models for PDE learning has shown promising advances. However, a persistent trade-off between execution success rate and numerical accuracy arises, particularly when generalization to unseen parameters and boundary conditions is required. In this work, we propose OpInf-LLM, an LLM parametric PDE solving framework based on operator inference. The proposed framework leverages a small amount of solution data to enable accurate prediction of diverse PDE instances, including unseen parameters and configurations, and provides seamless integration with LLMs for natural language specification of PDE solving tasks. Its low computational demands and unified tool interface further enable a high execution success rate across heterogeneous settings. By combining operator inference with LLM capabilities, OpInf-LLM opens new possibilities for generalizable reduced-order modeling in LLM-based PDE solving.

Xiaoyu Xie, Saviz Mowlavi, Mouhacine Benosman

Spatiotemporal modeling is critical for understanding complex systems across various scientific and engineering disciplines, but governing equations are often not fully known or computationally intractable due to inherent system complexity. Data-driven reduced-order models (ROMs) offer a promising approach for fast and accurate spatiotemporal forecasting by computing solutions in a compressed latent space. However, these models often neglect temporal correlations between consecutive snapshots when constructing the latent space, leading to suboptimal compression, jagged latent trajectories, and limited extrapolation ability over time. To address these issues, this paper introduces a continuous operator learning framework that incorporates jerk regularization into the learning of the compressed latent space. This jerk regularization promotes smoothness and sparsity of latent space dynamics, which not only yields enhanced accuracy and convergence speed but also helps identify intrinsic latent space coordinates. Consisting of an implicit neural representation (INR)-based autoencoder and a neural ODE latent dynamics model, the framework allows for inference at any desired spatial or temporal resolution. The effectiveness of this framework is demonstrated through a two-dimensional unsteady flow problem governed by the Navier-Stokes equations, highlighting its potential to expedite high-fidelity simulations in various scientific and engineering applications.

Xiangyuan Zhang, Saviz Mowlavi, Mouhacine Benosman et al.

Dimensionality reduction is crucial for controlling nonlinear partial differential equations (PDE) through a "reduce-then-design" strategy, which identifies a reduced-order model and then implements model-based control solutions. However, inaccuracies in the reduced-order modeling can substantially degrade controller performance, especially in PDEs with chaotic behavior. To address this issue, we augment the reduce-then-design procedure with a policy optimization (PO) step. The PO step fine-tunes the model-based controller to compensate for the modeling error from dimensionality reduction. This augmentation shifts the overall strategy into reduce-then-design-then-adapt, where the model-based controller serves as a warm start for PO. Specifically, we study the state-feedback tracking control of PDEs that aims to align the PDE state with a specific constant target subject to a linear-quadratic cost. Through extensive experiments, we show that a few iterations of PO can significantly improve the model-based controller performance. Our approach offers a cost-effective alternative to PDE control using end-to-end reinforcement learning.

Zhuoyuan Wang, Raffaele Romagnoli, Saviz Mowlavi et al. · cmu

Physics-informed machine learning offers a promising framework for solving complex partial differential equations (PDEs) by integrating observational data with governing physical laws. However, learning PDEs with varying parameters and changing initial conditions and boundary conditions (ICBCs) with theoretical guarantees remains an open challenge. In this paper, we propose physics-informed deep B-spline networks, a novel technique that approximates a family of PDEs with different parameters and ICBCs by learning B-spline control points through neural networks. The proposed B-spline representation reduces the learning task from predicting solution values over the entire domain to learning a compact set of control points, enforces strict compliance to initial and Dirichlet boundary conditions by construction, and enables analytical computation of derivatives for incorporating PDE residual losses. While existing approximation and generalization theories are not applicable in this setting - where solutions of parametrized PDE families are represented via B-spline bases - we fill this gap by showing that B-spline networks are universal approximators for such families under mild conditions. We also derive generalization error bounds for physics-informed learning in both elliptic and parabolic PDE settings, establishing new theoretical guarantees. Finally, we demonstrate in experiments that the proposed technique has improved efficiency-accuracy tradeoffs compared to existing techniques in a dynamical system problem with discontinuous ICBCs and can handle nonhomogeneous ICBCs and non-rectangular domains.

Jonah Botvinick-Greenhouse, Wael H. Ali, Mouhacine Benosman et al.

We introduce adaptive-basis physics-informed neural networks (AB-PINNs), a novel approach to domain decomposition for training PINNs in which existing subdomains dynamically adapt to the intrinsic features of the unknown solution. Drawing inspiration from classical mesh refinement techniques, we also modify the domain decomposition on-the-fly throughout training by introducing new subdomains in regions of high residual loss, thereby providing additional expressive power where the solution of the differential equation is challenging to represent. Our flexible approach to domain decomposition is well-suited for multiscale problems, as different subdomains can learn to capture different scales of the underlying solution. Moreover, the ability to introduce new subdomains during training helps prevent convergence to unwanted local minima and can reduce the need for extensive hyperparameter tuning compared to static domain decomposition approaches. Throughout, we present comprehensive numerical results which demonstrate the effectiveness of AB-PINNs at solving a variety of complex multiscale partial differential equations.