Symbolic regression aims to discover concise, interpretable mathematical expressions that satisfy desired objectives, such as fitting data, posing a highly combinatorial optimization problem. While genetic programming has been the dominant approach, recent efforts have explored reinforcement learning methods for improving search efficiency. Monte Carlo Tree Search (MCTS), with its ability to balance exploration and exploitation through guided search, has emerged as a promising technique for symbolic expression discovery. However, its traditional bandit strategies and sequential symbol construction often limit performance. In this work, we propose an improved MCTS framework for symbolic regression that addresses these limitations through two key innovations: (1) an extreme bandit allocation strategy tailored for identifying globally optimal expressions, with finite-time performance guarantees under polynomial reward decay assumptions; and (2) evolution-inspired state-jumping actions such as mutation and crossover, which enable non-local transitions to promising regions of the search space. These state-jumping actions also reshape the reward landscape during the search process, improving both robustness and efficiency. We conduct a thorough numerical study to the impact of these improvements and benchmark our approach against existing symbolic regression methods on a variety of datasets, including both ground-truth and black-box datasets. Our approach achieves competitive performance with state-of-the-art libraries in terms of recovery rate, attains favorable positions on the Pareto frontier of accuracy versus model complexity.

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., Geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting Geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our Geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Comprehensive cost-accuracy experiments show that Geo-FNO is $10^5$ times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.

Nonlinear dynamics are ubiquitous in science and engineering applications, but the physics of most complex systems is far from being fully understood. Discovering interpretable governing equations from measurement data can help us understand and predict the behavior of complex dynamic systems. Although extensive work has recently been done in this field, robustly distilling explicit model forms from very sparse data with considerable noise remains intractable. Moreover, quantifying and propagating the uncertainty of the identified system from noisy data is challenging, and relevant literature is still limited. To bridge this gap, we develop a novel Bayesian spline learning framework to identify parsimonious governing equations of nonlinear (spatio)temporal dynamics from sparse, noisy data with quantified uncertainty. The proposed method utilizes spline basis to handle the data scarcity and measurement noise, upon which a group of derivatives can be accurately computed to form a library of candidate model terms. The equation residuals are used to inform the spline learning in a Bayesian manner, where approximate Bayesian uncertainty calibration techniques are employed to approximate posterior distributions of the trainable parameters. To promote the sparsity, an iterative sequential-threshold Bayesian learning approach is developed, using the alternative direction optimization strategy to systematically approximate L0 sparsity constraints. The proposed algorithm is evaluated on multiple nonlinear dynamical systems governed by canonical ordinary and partial differential equations, and the merit/superiority of the proposed method is demonstrated by comparison with state-of-the-art methods.

In this paper, we study the power iteration algorithm for the asymmetric spiked tensor model, as introduced in Richard and Montanari (2014). We give necessary and sufficient conditions for the convergence of the power iteration algorithm. When the power iteration algorithm converges, for the rank one spiked tensor model, we show the estimators for the spike strength and linear functionals of the signal are asymptotically Gaussian; for the multi-rank spiked tensor model, we show the estimators are asymptotically mixtures of Gaussian. This new phenomenon is different from the spiked matrix model. Using these asymptotic results of our estimators, we construct valid and efficient confidence intervals for spike strengths and linear functionals of the signals.