# Sequential Physics-Constrained Neural Operator Forward Modeling for the textitNorne Reservoir System

arXiv: https://arxiv.org/abs/2605.28909 · HTML: https://mlsanity.com/paper/2605.28909/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 27 May 2026.

## Builds on (most cited first, 14 of 14)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Scheduled Sampling for Sequence Prediction with Recurrent Neural Networks](https://mlsanity.com/paper/1506.03099/) (2015)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Sequence Level Training with Recurrent Neural Networks](https://mlsanity.com/paper/1511.06732/) (2015)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Professor Forcing: A New Algorithm for Training Recurrent Networks](https://mlsanity.com/paper/1610.09038/) (2016)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [On the difficulty of learning chaotic dynamics with RNNs](https://mlsanity.com/paper/2110.07238/) (2021)
- [Unbiasing Truncated Backpropagation Through Time](https://mlsanity.com/paper/1705.08209/) (2017)

## Abstract (arXiv)

We develop a comprehensive mathematical and computational framework for sequential surrogate modeling of three-phase black-oil reservoir dynamics using neural operators, with particular emphasis on Fourier Neural Operators (FNO) and their physics-informed variant (PINO). The application focus is the Norne benchmark reservoir, defined on a heterogeneous 46×112×22 grid (N=113,344 cells), with a production history spanning T=30 timesteps covering 3298 days. Our theoretical contributions are organized around four interlocking problems: (1) functional-analytic formulation in a product-Sobolev-space setting, including well-posedness of the implicit timestep map and sharp local Lipschitz estimates; (2) covariate shift quantification, proving that the Wasserstein-2 distance grows as W2 ≤ varepsilon(Ln-1)/(L-1), with exponential population-risk discrepancy for L>1; (3) physics-constrained spectral stability, showing PINO training with λR ≥ λ^*R reduces the learned Jacobian spectral radius to ρF + CλR-1/2, yielding uniform-in-time rollout error |δn| ≤ varepsilon/(1-ρ); and (4) K-step TBPTT gradient analysis, deriving geometric bias decay O(ρK), optimal window K^ = O(log(T/σ2)), and Adam convergence O(1/√t) + O(ρK^*). Empirical validation confirms all theoretical predictions: autoregressive PINO surrogates sustain R2>0.99 (oil), R2>0.90 (gas), R2≈ 0.80 (pressure), and monotonically improving R2 (water) across the full 3298-day horizon, trained on eight NVIDIA B200 GPUs in under one hour. A 1000-member ensemble runs in under one minute on a single B200 GPU, giving a ~104× wall-clock speedup over the OPM finite-volume simulator.