# Conflict-Aware Harmonized Rotational Gradient for Multiscale Kinetic Regimes

arXiv: https://arxiv.org/abs/2604.24745 · HTML: https://mlsanity.com/paper/2604.24745/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 55 references that are in the map.
- First posted on arXiv 27 Apr 2026.

## Builds on (most cited first, 23 of 23)

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- [End-to-End Multi-Task Learning with Attention](https://mlsanity.com/paper/1803.10704/) (2018)
- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [Which Tasks Should Be Learned Together in Multi-task Learning?](https://mlsanity.com/paper/1905.07553/) (2019)
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- [Multi-Task Learning as a Bargaining Game](https://mlsanity.com/paper/2202.01017/) (2022)
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- [FAMO: Fast Adaptive Multitask Optimization](https://mlsanity.com/paper/2306.03792/) (2023)
- [Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective](https://mlsanity.com/paper/2502.00604/) (2025)
- [ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2408.11104/) (2024)
- [HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks](https://mlsanity.com/paper/2111.01008/) (2021)
- [Parameterized Physics-informed Neural Networks for Parameterized PDEs](https://mlsanity.com/paper/2408.09446/) (2024)
- [Meta-Auto-Decoder for Solving Parametric Partial Differential Equations](https://mlsanity.com/paper/2111.08823/) (2021)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networks](https://mlsanity.com/paper/2306.02816/) (2023)
- [Separable Physics-informed Neural Networks for Solving the BGK Model of the Boltzmann Equation](https://mlsanity.com/paper/2403.06342/) (2024)

## Abstract (arXiv)

In this paper, we propose a harmonized rotational gradient method, termed HRGrad, for simultaneously tackling multiscale time-dependent kinetic problems with varying small parameters. These parameters exhibit asymptotic transitions from microscopic to macroscopic physics, making it a challenging multi-task problem to solve over all ranges simultaneously. Solving tasks in different asymptotic regions often encounter gradient conflicts, which can lead to the failure of multi-task learning. To address this challenge, we explicitly encode a hidden representation of these parameters, ensuring that the corresponding solving tasks are serialized for simultaneous training. Furthermore, to mitigate gradient conflicts, we segment the prediction results to construct task losses and introduce a novel gradient alignment metric to ensure a positive dot product between the final update and each loss-specific gradient. This metric maintains consistent optimization rates for all task losses and dynamically adjusts gradient magnitudes based on conflict levels. Moreover, we provide a mathematical proof demonstrating the convergence of the HRGrad method, which is evaluated across a range of challenging asymptotic-preserving neural networks (APNNs) scenarios. We conduct an extensive set of experiments encompassing the Bhatnagar-Gross-Krook (BGK) equation and the linear transport equation in all ranges of Knudsen number. Our results indicate that HRGrad effectively overcomes the `failure modes' of APNNs in these problems.