# Parabolic Continual Learning

arXiv: https://arxiv.org/abs/2503.02117 · HTML: https://mlsanity.com/paper/2503.02117/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 25 references that are in the map.
- First posted on arXiv 3 Mar 2025.

## Builds on (most cited first, 18 of 18)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Learning without Forgetting](https://mlsanity.com/paper/1606.09282/) (2016)
- [Gradient Episodic Memory for Continual Learning](https://mlsanity.com/paper/1706.08840/) (2017)
- [Efficient Lifelong Learning with A-GEM](https://mlsanity.com/paper/1812.00420/) (2018)
- [Dark Experience for General Continual Learning: a Strong, Simple Baseline](https://mlsanity.com/paper/2004.07211/) (2020)
- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [Learning to Learn without Forgetting by Maximizing Transfer and Minimizing Interference](https://mlsanity.com/paper/1810.11910/) (2018)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Learning Fast, Learning Slow: A General Continual Learning Method based on Complementary Learning System](https://mlsanity.com/paper/2201.12604/) (2022)
- [CLiMB: A Continual Learning Benchmark for Vision-and-Language Tasks](https://mlsanity.com/paper/2206.09059/) (2022)
- [On Quadratic Penalties in Elastic Weight Consolidation](https://mlsanity.com/paper/1712.03847/) (2017)
- [Reproducibility Report: La-MAML: Look-ahead Meta Learning for Continual Learning](https://mlsanity.com/paper/2102.05824/) (2021)
- [The Ideal Continual Learner: An Agent That Never Forgets](https://mlsanity.com/paper/2305.00316/) (2023)
- [Distributional Hamilton-Jacobi-Bellman Equations for Continuous-Time Reinforcement Learning](https://mlsanity.com/paper/2205.12184/) (2022)
- [Meta Continual Learning via Dynamic Programming](https://mlsanity.com/paper/2008.02219/) (2020)
- [Navigating Memory Construction by Global Pseudo-Task Simulation for Continual Learning](https://mlsanity.com/paper/2210.08442/) (2022)
- [Base Models for Parabolic Partial Differential Equations](https://mlsanity.com/paper/2407.12234/) (2024)

## Built on by (most cited first, 2 of 2)

- [Elliptic Loss Regularization](https://mlsanity.com/paper/2503.02138/) (2025)
- [The Sequential Price of Continual Learning](https://mlsanity.com/paper/2609.29674/) (2026)

## Abstract (arXiv)

Regularizing continual learning techniques is important for anticipating algorithmic behavior under new realizations of data. We introduce a new approach to continual learning by imposing the properties of a parabolic partial differential equation (PDE) to regularize the expected behavior of the loss over time. This class of parabolic PDEs has a number of favorable properties that allow us to analyze the error incurred through forgetting and the error induced through generalization. Specifically, we do this through imposing boundary conditions where the boundary is given by a memory buffer. By using the memory buffer as a boundary, we can enforce long term dependencies by bounding the expected error by the boundary loss. Finally, we illustrate the empirical performance of the method on a series of continual learning tasks.