Adaptive control of reaction-diffusion PDEs via neural operator-approximated gain kernels

Neural operator approximations of the gain kernels in PDE backstepping has emerged as a viable method for implementing controllers in real time. With such an approach, one approximates the gain kernel, which maps the plant coefficient into the solution of a PDE, with a neural operator. It is in adaptive control that the benefit of the neural operator is realized, as the kernel PDE solution needs to be computed online, for every updated estimate of the plant coefficient. We extend the neural operator methodology from adaptive control of a hyperbolic PDE to adaptive control of a benchmark parabolic PDE (a reaction-diffusion equation with a spatially-varying and unknown reaction coefficient). We prove global stability and asymptotic regulation of the plant state for a Lyapunov design of parameter adaptation. The key technical challenge of the result is handling the 2D nature of the gain kernels and proving that the target system with two distinct sources of perturbation terms, due to the parameter estimation error and due to the neural approximation error, is Lyapunov stable. To verify our theoretical result, we present simulations achieving calculation speedups up to 45x relative to the traditional finite difference solvers for every timestep in the simulation trajectory.

Combining Neural Networks and Symbolic Regression for Analytical Lyapunov Function Discovery

We propose CoNSAL (Combining Neural networks and Symbolic regression for Analytical Lyapunov function) to construct analytical Lyapunov functions for nonlinear dynamic systems. This framework contains a neural Lyapunov function and a symbolic regression component, where symbolic regression is applied to distill the neural network to precise analytical forms. Our approach utilizes symbolic regression not only as a tool for translation but also as a means to uncover counterexamples. This procedure terminates when no counterexamples are found in the analytical formulation. Compared with previous results, our algorithm directly produces an analytical form of the Lyapunov function with improved interpretability in both the learning process and the final results. We apply our algorithm to 2-D inverted pendulum, path following, Van Der Pol Oscillator, 3-D trig dynamics, 4-D rotating wheel pendulum, 6-D 3-bus power system, and demonstrate that our algorithm successfully finds their valid Lyapunov functions.

PDE Control Gym: A Benchmark for Data-Driven Boundary Control of Partial Differential Equations

Over the last decade, data-driven methods have surged in popularity, emerging as valuable tools for control theory. As such, neural network approximations of control feedback laws, system dynamics, and even Lyapunov functions have attracted growing attention. With the ascent of learning based control, the need for accurate, fast, and easy-to-use benchmarks has increased. In this work, we present the first learning-based environment for boundary control of PDEs. In our benchmark, we introduce three foundational PDE problems - a 1D transport PDE, a 1D reaction-diffusion PDE, and a 2D Navier-Stokes PDE - whose solvers are bundled in an user-friendly reinforcement learning gym. With this gym, we then present the first set of model-free, reinforcement learning algorithms for solving this series of benchmark problems, achieving stability, although at a higher cost compared to model-based PDE backstepping. With the set of benchmark environments and detailed examples, this work significantly lowers the barrier to entry for learning-based PDE control - a topic largely unexplored by the data-driven control community. The entire benchmark is available on Github along with detailed documentation and the presented reinforcement learning models are open sourced.

Improving Sequential Market Clearing via Value-oriented Renewable Energy Forecasting

Large penetration of renewable energy sources (RESs) brings huge uncertainty into the electricity markets. While existing deterministic market clearing fails to accommodate the uncertainty, the recently proposed stochastic market clearing struggles to achieve desirable market properties. In this work, we propose a value-oriented forecasting approach, which tactically determines the RESs generation that enters the day-ahead market. With such a forecast, the existing deterministic market clearing framework can be maintained, and the day-ahead and real-time overall operation cost is reduced. At the training phase, the forecast model parameters are estimated to minimize expected day-ahead and real-time overall operation costs, instead of minimizing forecast errors in a statistical sense. Theoretically, we derive the exact form of the loss function for training the forecast model that aligns with such a goal. For market clearing modeled by linear programs, this loss function is a piecewise linear function. Additionally, we derive the analytical gradient of the loss function with respect to the forecast, which inspires an efficient training strategy. A numerical study shows our forecasts can bring significant benefits of the overall cost reduction to deterministic market clearing, compared to quality-oriented forecasting approach.

Adaptive Neural-Operator Backstepping Control of a Benchmark Hyperbolic PDE

To stabilize PDEs, feedback controllers require gain kernel functions, which are themselves governed by PDEs. Furthermore, these gain-kernel PDEs depend on the PDE plants' functional coefficients. The functional coefficients in PDE plants are often unknown. This requires an adaptive approach to PDE control, i.e., an estimation of the plant coefficients conducted concurrently with control, where a separate PDE for the gain kernel must be solved at each timestep upon the update in the plant coefficient function estimate. Solving a PDE at each timestep is computationally expensive and a barrier to the implementation of real-time adaptive control of PDEs. Recently, results in neural operator (NO) approximations of functional mappings have been introduced into PDE control, for replacing the computation of the gain kernel with a neural network that is trained, once offline, and reused in real-time for rapid solution of the PDEs. In this paper, we present the first result on applying NOs in adaptive PDE control, presented for a benchmark 1-D hyperbolic PDE with recirculation. We establish global stabilization via Lyapunov analysis, in the plant and parameter error states, and also present an alternative approach, via passive identifiers, which avoids the strong assumptions on kernel differentiability. We then present numerical simulations demonstrating stability and observe speedups up to three orders of magnitude, highlighting the real-time efficacy of neural operators in adaptive control. Our code (Github) is made publicly available for future researchers.

Moving-Horizon Estimators for Hyperbolic and Parabolic PDEs in 1-D

Observers for PDEs are themselves PDEs. Therefore, producing real time estimates with such observers is computationally burdensome. For both finite-dimensional and ODE systems, moving-horizon estimators (MHE) are operators whose output is the state estimate, while their inputs are the initial state estimate at the beginning of the horizon as well as the measured output and input signals over the moving time horizon. In this paper we introduce MHEs for PDEs which remove the need for a numerical solution of an observer PDE in real time. We accomplish this using the PDE backstepping method which, for certain classes of both hyperbolic and parabolic PDEs, produces moving-horizon state estimates explicitly. Precisely, to explicitly produce the state estimates, we employ a backstepping transformation of a hard-to-solve observer PDE into a target observer PDE, which is explicitly solvable. The MHEs we propose are not new observer designs but simply the explicit MHE realizations, over a moving horizon of arbitrary length, of the existing backstepping observers. Our PDE MHEs lack the optimality of the MHEs that arose as duals of MPC, but they are given explicitly, even for PDEs. In the paper we provide explicit formulae for MHEs for both hyperbolic and parabolic PDEs, as well as simulation results that illustrate theoretically guaranteed convergence of the MHEs.

GeoGalactica: A Scientific Large Language Model in Geoscience

Large language models (LLMs) have achieved huge success for their general knowledge and ability to solve a wide spectrum of tasks in natural language processing (NLP). Due to their impressive abilities, LLMs have shed light on potential inter-discipline applications to foster scientific discoveries of a specific domain by using artificial intelligence (AI for science, AI4S). In the meantime, utilizing NLP techniques in geoscience research and practice is wide and convoluted, contributing from knowledge extraction and document classification to question answering and knowledge discovery. In this work, we take the initial step to leverage LLM for science, through a rather straightforward approach. We try to specialize an LLM into geoscience, by further pre-training the model with a vast amount of texts in geoscience, as well as supervised fine-tuning (SFT) the resulting model with our custom collected instruction tuning dataset. These efforts result in a model GeoGalactica consisting of 30 billion parameters. To our best knowledge, it is the largest language model for the geoscience domain. More specifically, GeoGalactica is from further pre-training of Galactica. We train GeoGalactica over a geoscience-related text corpus containing 65 billion tokens curated from extensive data sources in the big science project Deep-time Digital Earth (DDE), preserving as the largest geoscience-specific text corpus. Then we fine-tune the model with 1 million pairs of instruction-tuning data consisting of questions that demand professional geoscience knowledge to answer. In this technical report, we will illustrate in detail all aspects of GeoGalactica, including data collection, data cleaning, base model selection, pre-training, SFT, and evaluation. We open-source our data curation tools and the checkpoints of GeoGalactica during the first 3/4 of pre-training.

Learning A Foundation Language Model for Geoscience Knowledge Understanding and Utilization

Large language models (LLMs)have achieved great success in general domains of natural language processing. In this paper, we bring LLMs to the realm of geoscience, with the objective of advancing research and applications in this field. To this end, we present the first-ever LLM in geoscience, K2, alongside a suite of resources developed to further promote LLM research within geoscience. For instance, we have curated the first geoscience instruction tuning dataset, GeoSignal, which aims to align LLM responses to geoscience-related user queries. Additionally, we have established the first geoscience benchmark, GeoBenchmark, to evaluate LLMs in the context of geoscience. In this work, we experiment with a complete recipe to adapt a pretrained general-domain LLM to the geoscience domain. Specifically, we further train the LLaMA-7B model on over 1 million pieces of geoscience literature and utilize GeoSignal's supervised data to fine-tune the model. Moreover, we share a protocol that can efficiently gather domain-specific data and construct domain-supervised data, even in situations where manpower is scarce. Experiments conducted on the GeoBenchmark demonstrate the the effectiveness of our approach and datasets.

Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs

Unlike ODEs, whose models involve system matrices and whose controllers involve vector or matrix gains, PDE models involve functions in those roles functional coefficients, dependent on the spatial variables, and gain functions dependent on space as well. The designs of gains for controllers and observers for PDEs, such as PDE backstepping, are mappings of system model functions into gain functions. These infinite dimensional nonlinear operators are given in an implicit form through PDEs, in spatial variables, which need to be solved to determine the gain function for each new functional coefficient of the PDE. The need for solving such PDEs can be eliminated by learning and approximating the said design mapping in the form of a neural operator. Learning the neural operator requires a sufficient number of prior solutions for the design PDEs, offline, as well as the training of the operator. In recent work, we developed the neural operators for PDE backstepping designs for first order hyperbolic PDEs. Here we extend this framework to the more complex class of parabolic PDEs. The key theoretical question is whether the controllers are still stabilizing, and whether the observers are still convergent, if they employ the approximate functional gains generated by the neural operator. We provide affirmative answers to these questions, namely, we prove stability in closed loop under gains produced by neural operators. We illustrate the theoretical results with numerical tests and publish our code on github. The neural operators are three orders of magnitude faster in generating gain functions than PDE solvers for such gain functions. This opens up the opportunity for the use of this neural operator methodology in adaptive control and in gain scheduling control for nonlinear PDEs.

Machine Learning Accelerated PDE Backstepping Observers

State estimation is important for a variety of tasks, from forecasting to substituting for unmeasured states in feedback controllers. Performing real-time state estimation for PDEs using provably and rapidly converging observers, such as those based on PDE backstepping, is computationally expensive and in many cases prohibitive. We propose a framework for accelerating PDE observer computations using learning-based approaches that are much faster while maintaining accuracy. In particular, we employ the recently-developed Fourier Neural Operator (FNO) to learn the functional mapping from the initial observer state and boundary measurements to the state estimate. By employing backstepping observer gains for previously-designed observers with particular convergence rate guarantees, we provide numerical experiments that evaluate the increased computational efficiency gained with FNO. We consider the state estimation for three benchmark PDE examples motivated by applications: first, for a reaction-diffusion (parabolic) PDE whose state is estimated with an exponential rate of convergence; second, for a parabolic PDE with exact prescribed-time estimation; and, third, for a pair of coupled first-order hyperbolic PDEs that modeling traffic flow density and velocity. The ML-accelerated observers trained on simulation data sets for these PDEs achieves up to three orders of magnitude improvement in computational speed compared to classical methods. This demonstrates the attractiveness of the ML-accelerated observers for real-time state estimation and control.