Michael McCabe, Payel Mukhopadhyay, Tanya Marwah, Bruno Régaldo-Saint Blancard, François Rozet, Cristiana Diaconu, Lucas Meyer, Kaze W. K. Wong, Hadi Sotoudeh, Alberto Bietti, Irina Espejo, Rio Fear, Siavash Golkar, Tom Hehir, Keiya Hirashima, Geraud Krawezik, François Lanusse, Rudy Morel, Ruben Ohana, Liam Parker, Mariel Pettee, Jeff Shen,
Kyunghyun Cho, Miles Cranmer, Shirley Ho
Paper: https://arxiv.org/abs/2511.15684
Code: https://github.com/PolymathicAI/walrus
Blog: https://polymathic-ai.org/blog/walrus/
Attribution for Walrus background goes to Getty images and is licensed via Unsplash+.
This project page is intended to show case simulation rollouts generated by Walrus, the new multi-domain foundation model for continuum dynamics released by Polymathic. It was trained on 19 different physical scenarios spanning 63 physical variables in both 2 and 3D. Architecturally, Walrus is fundamentally just a large factorized space/time transformer using an encoder-processor-decoder structure, but takes advantage of a few new tricks to improve rollout stability, flexibility, and training throughput. For the solutions seen below, the most relevant are:
Patch Jittering
Walrus suppressed the growth of long-run instabilities through the use of patch jittering. Patch jittering involves randomly translating the reference frame (with padding for boundaries) before each step. While the paper goes into more theoretical detail on why this works, the core idea is that the specific downsampling pattern leads to predictable accumulation of error and that randomizing this process can help alleviate this pathology.
Adaptive Compute
To handle varying compute budgets and problem complexities, we also employ stride modulation to allow users to adjust their downstream resolution. During pretraining, this was used to keep internal resolution fairly consistent (32/33 per dim in 2D, 16/17 in 3D). In this approach, the downsampling layers of the encoder/decoder will dynamically adjust their stride based on a target internal resolution.
Combining these two tools lets us produce the stable, highly accurate solutions we visualize below (either full 2D or central slices of 3D).
(Aspect ratio will look distorted until the video is played)
Acoustic Scattering (Discontinuous).mp4Source: The Well
Dimensionality: 2D
Description: Simple acoustic wave propogation over a domain split into two continuously varying sub-domains with a single discountinuous interface.
Simulation Details: Riemann solver from Clawpack without AMR enabled. Time integration using SSP104 integrator. Each simulation takes ~15 minutes on 64 Icelake cores.
Acoustic Scattering (Inclusions).mp4Source: The Well
Dimensionality: 2D
Description: Simple acoustic wave propogation over a domain split into two continuously varying sub-domains with a single discountinuous interface. With additive randomly generating inclusions (materials of significantly different density).
Simulation Details: Riemann solver from Clawpack without AMR. Time integration using SSP104 integrator. Each simulation takes ~15 minutes on 64 Icelake cores.
Acoustic Scattering (Maze).mp4Source: The Well
Dimensionality: 2D
Description: Simple acoustic wave propogation through maze-like structures.
Simulation Details: Riemann solver from Clawpack without AMR. Time integration using SSP104 integrator. Each simulation takes ~20 minutes on 64 Icelake cores.
Active Matter.mp4Source: The Well
Dimensionality: 2D
Description: Simulation of a continuum theory describing the dynamics of N rod-like active particles immersed in a Stokes fluid. Used to model crystalline flows in materials or bacterial flows in biology.
Simulation Details: Fourier pseudospectral discretization of periodic box. SBDF2 time integrator where the linear terms are handled implicitly and the nonlinear terms explicitly. Each simulation takes ~20 minutes on an A100 80GB GPU in fp64 precision
Compressible Navier-Stokes (Rand).mp4Source: PDEBench
Dimensionality: 3D
Description: Simulation of mixed Mach number compressible Navier-Stokes on periodic cube with random sinusoidal initial conditions.
Simulation Details: The numerical solution was calculated with the temporally and spatially 2nd-order HLLC scheme with the MUSCL method for the inviscid part, and the central difference scheme for the viscous part.
Compressible Navier-Stokes (Turbulence).mp4Source: PDEBench
Dimensionality: 3D
Description: Simulation of compressible Navier-Stokes on periodic cube at M=1 with uniform initial mass, density, and pressure and turbulent velocity.
Simulation Details: The numerical solution was calculated with the temporally and spatially 2nd-order HLLC scheme with the MUSCL method for the inviscid part, and the central difference scheme for the viscous part.
Convective Envelope of Red Supergiant Star.mp4Source: The Well
Dimensionality: 3D
Description: Hydrodynamic simulations of the convective envelope of red supergiant stars.
Simulation Details: Log-spherical (log r) domain using Godunov method in Athena++. Ghost zones to enforce BCs. The simulations were generated during 2 months on 80 nodes of NASA Pleiades Skylake CPU nodes.
Euler Multiquarants (Open BC).mp4Source: The Well
Dimensionality: 2D
Description: Evolution of different gases starting from piecewise-constant initial conditions in each quadrant, simulated under open boundary conditions.
Simulation Details: Flux-limited (MC) Lax-Wendroff discretization in Clawpack. On averages, each simulation took ~5 minutes on 160 Icelake cores.
Euler Multiquarants (Periodic BC).mp4Source: The Well
Dimensionality: 2D
Description: Evolution of different gases starting from piecewise-constant initial conditions in each quadrant, simulated under periodic boundary conditions.
Simulation Details: Flux-limited (MC) Lax-Wendroff discretization in Clawpack. On average, each simulation took ~5 minutes on 160 Icelake cores.
Flow Past an Object (Harmonics).mp4Source: Flowbench
Dimensionality: 2D
Description: Incompressible fluid flow past a stationary object, capturing vortex shedding and wake evolution, where the obstacle shape is a smooth parametric form generated from spherical-harmonic modes.
Simulation Details: LES formulation defined on geometry generated by tree-based tool solved using Dendro-KT and downsampled in ParaView.
Flow Past an Object (SkelNetOn).mp4Source: Flowbench
Dimensionality: 2D
Description: Incompressible fluid flow past a stationary object, capturing vortex shedding and wake evolution, where the obstacle shape is a non-parametric silhouette sampled from the SkelNetOn 2019 dataset.
Simulation Details: LES formulation defined on geometry generated by tree-based tool solved using Dendro-KT and downsampled in ParaView.
Gray-Scott Reaction-Diffusion.mp4Source: The Well
Dimensionality: 2D
Description: Simulation of the Gray–Scott reaction–diffusion system showing how two chemical species interact to form stable and dynamic patterns under different parameter regimes.
Simulation Details: Fourier spectral method implemented in Chebfun. Uses ETDRK4 integrator in time. Linear diffusion terms treated implicitly and the nonlinear reaction terms treated explicitly. Each trajectory takes ~4 minutes to generate on 40 Icelake cores.
Helmholtz Staircase.mp4Source: The Well
Dimensionality: 2D
Description: High-order accurate solution of acoustic scattering from a nonperiodic source by a periodic surface, relevant for its use in waveguide applications
Simulation Details: Uses the Floquet--Bloch transform with a high-order boundary integral equation (BIE) method to solve each of the quasiperiodic BVPs. Each trajectory takes ~6.6 minutes on 64 CPU cores.
Magnetohydrodynamics.mp4Source: The Well
Dimensionality: 3D
Description: Ideal Magnetohydrodynamics compressible turbulence, an essential component of the solar wind, galaxy formation, and of interstellar medium (ISM) dynamics drawn from CATS.
Simulation Details: Solved used third-order-accurate hybrid essentially non-oscillatory (ENO) scheme at 256^3 resolution with periodic boundaries and downsampled in frequency domain to 64^3. Wall time is ~48 hours per trajectory on 64 CPU cores.
Rayleigh–Bénard Convection.mp4Source: The Well
Dimensionality: 2D
Description: A horizontal layer of fluid heated from below and cooled from above, where buoyancy-driven motion causes warm fluid to rise and cool fluid to sink, forming convective cells and complex flow patterns.
Simulation Details: Mixed basis pseudospectral method in Dedalus on spatially oversampled (3/2) grid. IMEX RK222 in time. Wall time is ~10 minutes per trajectory at the lower range Rayleigh numbers and increases at higher Rayleigh number.
Richtmyer-Meshkov Problem.mp4Source: PDEGym
Dimensionality: 2D
Description: A shock-driven mixing problem where compressible flow described by the Euler equations evolves under the Richtmyer–Meshkov Instability with a passive tracer in a unit-square domain.
Simulation Details: Solved using FISH hydrodynamic code which provides high resolution finite-volume schemes.
Shear Flow.mp4Source: The Well
Dimensionality: 2D
Description: A two-dimensional incompressible shear flow with periodic boundary conditions, where adjacent fluid layers slide past each other and a passive tracer evolves under the resulting velocity field.
Simulation Details: Fourier pseudospectral method in Dedalus on spatially oversampled (3/2) grid. SBDF3 in time. Average wall time of ~2 minutes per trajectory.
Turbulence Gravity Cooling.mp4Source: The Well
Dimensionality: 3D
Description: Turbulence in an interstellar medium in various evolution stages of galaxies.
Simulation Details: Simulations are implemented with N-body/SPH code ASURA-FDPS A Density-Independent Smoothed Particle Hydrodynamics (DISPH) is employed to solve the hydrodynamic interaction. Simulated in ~222 node-hours per trajectory.
Turbulent Radiative Layer.mp4Source: The Well
Dimensionality: 2D
Description: An astrophysical radiative-convective layer where buoyancy-driven turbulence interacts with radiative cooling, producing complex coupled flow–temperature dynamics.
Simulation Details: Solved using Athena++ with AMR. Each simulation took ~48 seconds of wall-time on 40 CPU cores.
Viscoelastic Instability.mp4Source: The Well
Dimensionality: 2D
Description: A viscoelastic fluid in a channel whose flow becomes unstable because the fluid can stretch and store elastic stress, leading to complex, fluctuating motion.
Simulation Details: The data is generated using Dedalus codebase. Each trajectory takes ~24 hours on 32 CPU cores.