# FlowAdam: Implicit Regularization via Geometry-Aware Soft Momentum Injection

arXiv: https://arxiv.org/abs/2604.06652 · HTML: https://mlsanity.com/paper/2604.06652/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 24 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Lookahead Optimizer: k steps forward, 1 step back](https://mlsanity.com/paper/1907.08610/) (2019)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [ADAHESSIAN: An Adaptive Second Order Optimizer for Machine Learning](https://mlsanity.com/paper/2006.00719/) (2020)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [On the SDEs and Scaling Rules for Adaptive Gradient Algorithms](https://mlsanity.com/paper/2205.10287/) (2022)
- [Learning by solving differential equations](https://mlsanity.com/paper/2505.13397/) (2025)

## Built on by (most cited first, 1 of 1)

- [The Loss Does Not See the Basis, but Adam Does](https://mlsanity.com/paper/2608.05136/) (2026)

## Abstract (arXiv)

Adaptive moment methods such as Adam use a diagonal, coordinate-wise preconditioner based on exponential moving averages of squared gradients. This diagonal scaling is coordinate-system dependent and can struggle with dense or rotated parameter couplings, including those in matrix factorization, tensor decomposition, and graph neural networks, because it treats each parameter independently. We introduce FlowAdam, a hybrid optimizer that augments Adam with continuous gradient-flow integration via an ordinary differential equation (ODE). When EMA-based statistics detect landscape difficulty, FlowAdam switches to clipped ODE integration. Our central contribution is Soft Momentum Injection, which blends ODE velocity with Adam's momentum during mode transitions. This prevents the training collapse observed with naive hybrid approaches. Across coupled optimization benchmarks, the ODE integration provides implicit regularization, reducing held-out error by 10-22% on low-rank matrix/tensor recovery and 6% on Jester (real-world collaborative filtering), also surpassing tuned Lion and AdaBelief, while matching Adam on well-conditioned workloads (CIFAR-10). MovieLens-100K confirms benefits arise specifically from coupled parameter interactions rather than bias estimation. Ablation studies show that soft injection is essential, as hard replacement reduces accuracy from 100% to 82.5%.