# Hierarchical Implicit Neural Emulators

arXiv: https://arxiv.org/abs/2506.04528 · HTML: https://mlsanity.com/paper/2506.04528/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 55 references that are in the map.
- First posted on arXiv 5 Jun 2025.

## Builds on (most cited first, 30 of 32)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Feature Pyramid Networks for Object Detection](https://mlsanity.com/paper/1612.03144/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Multiscale Vision Transformers](https://mlsanity.com/paper/2104.11227/) (2021)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Neural General Circulation Models for Weather and Climate](https://mlsanity.com/paper/2311.07222/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [GenCast: Diffusion-based ensemble forecasting for medium-range weather](https://mlsanity.com/paper/2312.15796/) (2023)
- [DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecasting](https://mlsanity.com/paper/2306.01984/) (2023)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [ACE: A fast, skillful learned global atmospheric model for climate prediction](https://mlsanity.com/paper/2310.02074/) (2023)
- [Training neural operators to preserve invariant measures of chaotic attractors](https://mlsanity.com/paper/2306.01187/) (2023)
- [Learning to Predict the Cosmological Structure Formation](https://mlsanity.com/paper/1811.06533/) (2018)
- [Challenges of learning multi-scale dynamics with AI weather models: Implications for stability and one solution](https://mlsanity.com/paper/2304.07029/) (2023)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Differentiability in Unrolled Training of Neural Physics Simulators on Transient Dynamics](https://mlsanity.com/paper/2402.12971/) (2024)
- [Graph coarsening: From scientific computing to machine learning](https://mlsanity.com/paper/2106.11863/) (2021)
- [Thermalizer: Stable autoregressive neural emulation of spatiotemporal chaos](https://mlsanity.com/paper/2503.18731/) (2025)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [Field Level Neural Network Emulator for Cosmological N-body Simulations](https://mlsanity.com/paper/2206.04594/) (2022)
- [Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challenges](https://mlsanity.com/paper/2306.05014/) (2023)
- [Multigrid Neural Memory](https://mlsanity.com/paper/1906.05948/) (2019)
- [Embed and Emulate: Contrastive representations for simulation-based inference](https://mlsanity.com/paper/2409.18402/) (2024)
- [MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation](https://mlsanity.com/paper/2501.15987/) (2025)

## Built on by (most cited first, 10 of 10)

- [Project and Generate: Divergence-Free Neural Operators for Incompressible Flows](https://mlsanity.com/paper/2603.24500/) (2026)
- [Probing forced responses and causality in data-driven climate emulators: conceptual limitations and the role of reduced-order models](https://mlsanity.com/paper/2506.22552/) (2025)
- [Method of Manufactured Learning for Solver-free Training of Neural Operators](https://mlsanity.com/paper/2511.12890/) (2025)
- [Cheap2Rich: A Multi-Fidelity Framework for Data Assimilation and System Identification of Multiscale Physics -- Rotating Detonation Engines](https://mlsanity.com/paper/2601.20295/) (2026)
- [SENDAI: A Hierarchical Sparse-measurement, EfficieNt Data AssImilation Framework](https://mlsanity.com/paper/2601.21664/) (2026)
- [Accelerating PDE Surrogates via RL-Guided Mesh Optimization](https://mlsanity.com/paper/2603.02066/) (2026)
- [Nested Spatio-Temporal Time Series Forecasting](https://mlsanity.com/paper/2605.16447/) (2026)
- [Eigenanalysis framework for autoregressive neural emulators of multi-scale chaotic dynamics](https://mlsanity.com/paper/2608.16084/) (2026)
- [CasEm: A Cascade Architecture for Long-Horizon Neural Emulation](https://mlsanity.com/paper/2609.34246/) (2026)
- [ReAl: Refine neural operators via aligning dynamical flow](https://mlsanity.com/paper/2610.20369/) (2026)

## Abstract (arXiv)

Neural PDE solvers offer a powerful tool for modeling complex dynamical systems, but often struggle with error accumulation over long time horizons and maintaining stability and physical consistency. We introduce a multiscale implicit neural emulator that enhances long-term prediction accuracy by conditioning on a hierarchy of lower-dimensional future state representations. Drawing inspiration from the stability properties of numerical implicit time-stepping methods, our approach leverages predictions several steps ahead in time at increasing compression rates for next-timestep refinements. By actively adjusting the temporal downsampling ratios, our design enables the model to capture dynamics across multiple granularities and enforce long-range temporal coherence. Experiments on turbulent fluid dynamics show that our method achieves high short-term accuracy and produces long-term stable forecasts, significantly outperforming autoregressive baselines while adding minimal computational overhead.