**Thursday** at **1:50-2:45 PM** (unless otherwise noted)

For questions, contact Dr. Maria Han Veiga, Dr. Yulong Xing or Dr. Dongbin Xiu, Email: hanveiga dot 1@osu.edu, xing dot 205@osu.edu or xiu dot 16@osu.edu

- August 31
**François Ged**

Title: Matryoshka policy gradient for max-entropy reinforcement learning- November 16
**Jonathan Q Weare**

Title: Iterative random sparsification and convergence of a fast randomized Jacobi iteration- January 11
**Steven Wise**

Title: A Non-Isothermal Phase Field Crystal Model with Lattice Expansion- February 29
**Eirik Endeve**

Title: Application of Euler Equations to Model Astrophysical Flows- March 22
**Qingguo Hong**

Title: On the Activation Function Dependence of the Spectral Bias of Neural Networks- March 28
**Mrinal Kumar**

Title: TBD- April 12
**Deep Ray**

Title: Learning WENO for entropy stable schemes to solve conservation laws

DATE and TIME |
Location |
SPEAKER |
TITLE |

August 31 Thursday, 1:50pm |
In person Math Tower 154 |
François Ged (EPFL) |
Matryoshka policy gradient for max-entropy reinforcement learning |

November 16 Thursday, 1:50pm |
In person Math Tower 154 |
Johnathan Weare (NYU) |
Iterative random sparsification and convergence of a fast randomized Jacobi iteration |

January 11 Thursday, 1:50pm |
In person University Hall 0090 |
Steven Wise (Univ of Tennessee) |
A non-isothermal phase field crystal model with lattice expansion |

February 29 Thursday, 1:50pm |
In person Math Tower 100A |
Eirik Endeve (ORNL) |
Application of Euler equations to model astrophysical flows |

March 22 Friday, 4:00pm |
In person Math Tower 154 |
Qingguo Hong (Missouri Univ. of Science and Technology) |
On the Activation Function Dependence of the Spectral Bias of Neural Networks |

April 4 Thursday, 10:20am |
In person TBD |
Mrinal Kumar (OSU) |
TBD |

April 12 Friday, 4:10pm |
In person TBD |
Deep Ray (Univ of Maryland) |
Learning WENO for entropy stable schemes to solve conservation laws |

Reinforcement Learning (RL) is the area of Machine Learning addressing tasks where an agent interacts with its environment through a sequence of actions, chosen according to its policy. The agent’s goal is to maximize the rewards collected along the way, and in this talk, entropy bonuses are added to the rewards. This regularization technique has become more common, with benefits such as: enhancement of the exploration of the environment, uniqueness and stochasticity of the optimal policy, and more robustness of the agent to adversarial modifications of the rewards. Policy gradient algorithms are well suited to deal with large (possibly infinite) state and action spaces but theoretical guarantees have been lacking or obtained in rather specific settings. The case of infinite (continuous) state and action spaces remains mostly unsolved. In this talk, I will present a novel algorithm called Matryoshka Policy Gradient (MPG) that is both very intuitive and mathematically tractable. It uses so-called softmax policies and relies on the following idea: by fixing in advance a maximal horizon N, the agent with MPG learns to optimize policies for all smaller horizons simultaneously, that is from 1 to N, in a nested way (recalling the image of Matryoshka dolls). Theoretically, under mild assumptions, our most important results can be summarized as follows: 1. training converges to the unique optimum when the optimum belongs to the parametric space; 2. training converges to an explicit orthogonal projection of the unique optimum when it does not belong to the parametric space, this projection being optimal within that space; 3. for policies parametrized by a neural network, we provide a simple sufficient criterion at convergence for the global optimality of the limit, in terms of the neural tangent kernel of the neural network. Most notably, these convergence guarantees hold for infinite continuous state and action spaces. Numerically, we confirm the potential of our algorithm by successfully training an agent on two basic standard benchmarks from Open AI Gym, namely, frozen lake and cart pole. No background in RL is needed to understand the talk. Based on joint work with Prof. Maria Han Veiga.

The traditional methods of numerical linear algebra are prohibitively expensive for high-dimensional problems for which even a single matrix multiplication by a dense vector may be too costly. In this talk I will discuss a general framework for reducing the cost of classical iterative numerical linear algebra schemes by randomly sparsifying the approximate solution at each iteration. In the specific case of Jacobi iteration I will provide a thorough characterization of the randomized scheme's error properties. The talk is based on joint work with Tim Berkelbach, Sam Greene, Lek-Heng Lim, James Smith, and Rob Webber.

The phase field crystal modeling framework describes materials at atomic space scales on diffusive time scales. It has been used to study grain growth, fracture, crystallization, and other phenomena. In this talk I will describe some recent work with collaborators developing a thermo- dynamically consistent phase field crystal model that includes heat transport and lattice expansion and contraction. We use the theory of non-equilibrium thermodynamics, a formalism developed by Alt and Pawlow, and Onsager’s principle to give consistent laws of entropy production, and mass and energy conservation. I will show some preliminary numerical simulation results involving heat transport during solidification, and I will discuss some ideas on developing entropy and energy stable numerical methods.

The Euler equations are a key component of multi-physics models of many astrophysical systems, including core-collapse supernovae and binary neutron star mergers. While the Euler equations alone do not provide a realistic description of these systems, they can sometimes be used to study some of their aspects that are intractable with full-physics models. The study of the so-called standing accretion shock instability (SASI), which operates in a stalled supernova shock wave, and was discovered using idealized models based on the Euler equations, is a prime example. In this talk, I will first briefly discuss the Euler equations and a discontinuous Galerkin method to solve them numerically. Then I will discuss the application of the Euler equations to model the SASI. I will conclude by showing results from a recent study (Dunham et al., arXiv:2307.10904) comparing simulations using relativistic and non-relativistic implementations of the Euler equations.

Neural networks are universal function approximators which are known to generalize well despite being dramatically overparameterized. We study this phenomenon from the point of view of the spectral bias of neural networks. We provide a theoretical explanation for the spectral bias of ReLU neural networks by leveraging connections with the theory of finite element methods. Based upon this theory we predict that switching the activation function to a piecewise linear B-spline, namely the Hat function, will remove this spectral bias, which we verify empirically in a variety of settings. Our empirical studies also show that neural networks with the Hat activation function are trained significantly faster using stochastic gradient descent and ADAM. Combined with previous work showing that the Hat activation function also improves generalization accuracy on image classification tasks, this indicates that using the Hat activation provides significant advantages over the ReLU on certain problems.

Entropy stable solvers for hyperbolic conservation laws ensure the selection of a physically relevant (weak) solution of the underlying PDE. Among such methods, the TeCNO schemes [Fjordholm et al, 2012] form a class of high-order finite difference-based solvers that utilize reconstruction algorithms satisfying a critical “sign-property” at the cell-interfaces. However, only a handful of existing reconstructions are known to satisfy this property. In [Fjordholm-Ray, 2016], the first weighted essentially non-oscillatory (WENO) reconstruction satisfying the sign- property was developed. However, despite leading to provably entropy stable schemes, the numerical solutions using this reconstruction suffered from large under/overshoots near discontinuities.

In this talk, we propose an alternate approach to constructing WENO schemes possessing the sign-property. In particular, we train a neural network to determine the polynomial weights of the WENO scheme, while strongly constraining the network to satisfy the sign-property. The training data comprises smooth and discontinuous data that represent the local solution features of conservation laws. Additional constraints are built into the network to guarantee the expected order of convergence (for smooth solutions) with mesh refinement. We present several numerical results to demonstrate a significant improvement over the existing variants of WENO with the sign-property.

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