## Chair: Eric Vanden-Eijnden (NYU)

### Time: August 19th, 1:10pm-2pm ET, 19:10-20:00 CET, 01:10-02:00 GMT+8

**Implicit form neural network for learning scalar hyperbolic conservation laws**, Xiaoping Zhang (Wuhan University); Tao Cheng (Wuhan University); Lili Ju (University of South Carolina).

*Paper Highlight, by Yibo Yang*:

This paper proposes an unsupervised learning method —- Implicit Form Neural Networks (IFNN) using neural networks to solve partial differential equations (PDEs) whose solutions have discontinuities such as shock waves, etc. This approach is interesting as it leverages the conservation laws (prior knowledge) in the implicit form of PDEs solutions with neural networks to approximate the solution of differential equations. It also addresses problems (inviscid Burgers equations and Lighthill-Whitham-Richards model) that are not trivial for conventional physics-informed machine learning tools. Furthermore, the proposed framework is easy to implement and has potential scalabilities.

**Numerical Calabi-Yau metrics from holomorphic networks**, Michael R Douglas (Stony Brook University), Yidi Qi (Stony Brook University)

*Paper Highlight, by Lara Anderson*:

Calabi-Yau manifolds and their associated Ricci-flat metrics have played a crucial role within string theory for several decades already. These metrics determine much of the structure of the low energy physical theories and as such are essential for a detailed understanding of many string vacua. Although Yau’s theorem guarantees the existence of Ricci-flat metrics on these SU(3) holonomy spaces, the absence of continuous isometries has meant that no (non-trivial) examples have yet been found explicitly. As a result, the question of whether such metrics can be accurately approximated numerically is a very important one. Substantial work has occurred in recent years on this front, including implementations of the well-known Donaldson Algorithm (including work by one of the authors of the present paper).In this context, the question of whether machine learning/holomorphic networks can efficiently and accurately approximate Calabi-Yau metrics is a very interesting one. The present work provides an excellent ‘proof of principle’ of this approach and will be likely one of the first such papers in a rapidly emerging sub-field.

**A Data Driven Method for Computing Quasipotentials**, Bo Lin (National University of Singapore), Qianxiao Li (National University of Singapore); Weiqing Ren (National University of Singapore)

*Paper Highlight, by Alessandro Laio*:

The quasipotential plays an extremely important role in dynamic system theory: it can be seen as a generalization of the free energy to the case of dynamics which do not admit a zero-current equilibrium measure. Estimating it is a challenge inheriting all the difficulties of standard free energy estimates: “interesting” dynamic systems are typically multidimensional, bringing to a potential arbitrarity in the choice of the variables used to estimate and represent it, and are often characterized by metastability and kinetic traps, which make sampling difficult. The critical extra difficulty of estimating the quasipotential is the presence of currents, which induce systematic errors if one blindly uses an ordinary free energy estimator. The paper proposes to estimate the quasipotential V using a decomposition of the vector field generating the dynamics in a gradient field \grad V and in a vector field g orthogonal to \grad V. The key idea is then estimating V and g by two neural networks. This allows exploiting the representation power of neural networks in solving a difficult decomposition task. The approach is potentially very powerful, as illustrated in particular by the example of the Brusselator. A challenge for the future will be exploiting this idea to estimate the quasipotential in dynamics systems with a relevance in application, such as climate models, or molecular dynamics in the presence of non-conservative forces.

**Optimal Policies for a Pandemic: A Stochastic Game Approach and a Deep Learning Algorithm**, Yao Xuan (University of California, Santa Barbara); Robert A Balkin (Univeristy of California Santa Barbara); Jiequn Han (Princeton University); Ruimeng Hu (University of California, Santa Barbara), Hector Ceniceros (University of California, Santa Barbara)

*Paper Highlight, by Antoine Baker*:

The authors propose to model the pandemic policies as a stochastic game, where each region can control the lockdown or the effort put in the healthcare system. The optimal policies are obtained by the usual Hamilton-Jacobi-Bellman equations but these are numerically intractable as the number of regions grows. The deep fictitious play algorithm tackles this kind of problem by approximating the value function and the optimal controls by neural networks. I especially like the pedagogical presentation of the algorithm, which is accessible even to non-specialists of game theory.

**Reconstruction of Pairwise Interactions using Energy-Based Models**, Christoph Feinauer (Bocconi University), Carlo Lucibello (Bocconi University)

*Paper Highlight, by Manon Michel*:

This work deals with the determination of the pairwise couplings between binary variables while in the presence of higher-order interaction terms. The main point is to enforce a pairwise component in the energy function in an energy-based models (EBM) learnt over a multilayer perceptron architecture, and to reconstruct the complete pairwise couplings (thanks to the symmetry trick of the binary variables {-1,+1}), once training is over. Numerical experiments are carried over systems with Gaussian couplings. At the time of writing this text, questions still remain on the actual novelty of the approach on one hand, and on the efficiency on real applications on the other. Clarifications from the authors and discussions with other MSML participants during the presentation may then point to interesting developments. Namely, Regarding novelty, one could argue the general strategy of combining a simple and physical model with a universal approximator has been already proposed and applied. The presented work here should be then put into this perspective, although could argue the systems of interest (inverse Ising problems) and the reconstruction trick (known beforehand) are a new addition.