Shallow Water Equations - RDycore: A River / Flooding dynamical core for E3SM

Shallow water equations

The shallow-water equations (SWE) are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid. The shallow-water equations in unidirectional form are also called (de) Saint-Venant equations, after Adhémar Jean Claude Barré de Saint-Venant.

tristanbuckmaster (@tristanbuckmaster@mastodon.social)

Today, Levent Alpöge and I have made public three results: finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3d incompressible Euler. https://cims.nyu.edu/~tristanb/statement.pdf https://cims.nyu.edu/~tristanb/euler.pdf https://cims.nyu.edu/~tristanb/ipm.pdf https://cims.nyu.edu/~tristanb/boussinesq.pdf https://github.com/tristanbuckmaster/fluid_lean

lisyarus/webgpu-shallow-water

GPU shallow water solver using the virtual pipes model using WebGPU

Navier-Stokes Equation - Clay Mathematics Institute

This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

Adding Realistic Rivers to Random Terrain

Discusses a method for adding rivers to randomly generated terrain that adds a realistic touch.

Docs | Graze Turbostream

Details about the turbostream, a hydrated-reference variation of the jetstream

River AI — Intelligence that flows with you

River is building a new stack for Personal AI.

Mineral Water Atlas of the World

Navier-Stokes lost in translation: Why Lean verification of AI...

Autoformalisation is increasingly used to verify mathematical texts, including those generated by AI, as in OpenAI's announced proof of blow-up of solutions to the Navier-Stokes equations. In this...

GitHub - openai/NavierStokesAndEuler at 8937a8f4cbc7abaab5e9e97d1cc7f5d2319d9538

Lean certificates accompanying Navier-Stokes and Euler results - openai/NavierStokesAndEuler

Comparator | Reservoir

Reservoir is the package registry for Lake, the build system and package manager of the Lean programming language and theorem prover.

OpenAI solves Navier–Stokes, but research misconduct concerns remain | Olga Holtz posted on the topic | LinkedIn

OpenAI says its AI has solved a Millennium Prize Problem: Navier–Stokes. A human mathematician asks whether OpenAI committed research misconduct. As a mathematician working at the intersection of mathematics and AI, I find this truly messy while also potentially historic. Three things deserve attention. 1. The fine print IS the mathematics. The claimed construction starts with a three-dimensional incompressible fluid at rest. Under a smooth external force, its velocity becomes unbounded in finite time, even while its kinetic energy stays bounded. Crucially, the force itself remains smooth through the singularity. Forcing is explicitly permitted in Clay’s official breakdown alternatives C and D. Its presence does not disqualify the result. But this construction does not establish blow-up for unforced Navier–Stokes. That distinction matters. 2. This was a research system, not one clever prompt. OpenAI reports roughly 10,000 concurrent agents in the successful group and a result after about 88 hours. The Navier–Stokes effort used approximately 130 billion output tokens; Lean formalization and verification took another 17 hours. Human researchers redirected the effort and used tools to consolidate intermediate insights. The salient part here, as I already wrote earlier, is a new organization of mathematical discovery using an enormous amount of compute, not a faster chatbot. The published Lean formalization is important: there is something concrete for others to check. Let‘s see independent scrutiny of both the proof and whether the formal statement matches the intended theorem. 3. Proof verification is not provenance verification. The result follows the line of attack designed by Spanish mathematicians Diego Córdoba and Luis Martínez Zoroa, who appeared to be close to solving the problem themselves. Tristan Buckmaster has raised concerns about research priority and possible use of private Codex material (his statement is linked to the comment by Guillermo Núñez Ponasso below). Those concerns are not automatic proof of misconduct, yet: OpenAI recognizes Buckmaster and Levent Alpöge’s priority on forced Euler, a related but different problem. It denies accessing their unpublished work. But it also cannot rule out de-identified data from their OpenAI product usage helping improve its models, though it calls that unlikely. Is it though? That ‚leakage’ is quite possible, in my view. If correct, this is an extraordinary result. Let’s get independent verification. And let’s protect unpublished research — but exactly how to protect it from an AI lab if using its tools for this research? Recall that a verified theorem can settle a mathematical question. It cannot, by itself, settle scientific credit or establish that the discovery process was fair. Are you comfortable using an AI lab’s tools when that lab could also become your research competitor? https://lnkd.in/g4HzzUNp #AI #Mathematics #AIforScience #MillenniumProblems #NavierStokes | 54 comments on LinkedIn

Boiling The Ocean 5 - HedgeDoc