Geometry Conflict: Explaining and Controlling Forgetting in LLM Continual Post-Training

Continual post-training aims to extend large language models (LLMs) with new knowledge, skills, and behaviors, yet it remains unclear when sequential updates enable capability transfer and when they cause catastrophic forgetting. Existing methods mitigate forgetting through sequential fine-tuning, replay, regularization, or model merging, but offer limited criteria for determining when incorporating new updates is beneficial or harmful. In this work, we study LLM continual post-training through three questions: What drives forgetting? When do sequentially acquired capabilities transfer or interfere? How can compatibility be used to control update integration? We address these questions through task geometry: we represent each post-training task by its parameter update and study the covariance geometry induced by the update. Our central finding is that: forgetting can be considered as a state-relative update-integration failure, it arises when the covariance geometries induced by tasks misalign with the geometry of the evolving model state. Sequential updates transfer when they remain compatible with the model state shaped by previous updates, and interfere when state-relative geometry conflict becomes high. Motivated by this finding, we propose Geometry-Conflict Wasserstein Merging (GCWM), a data-free update-integration method that constructs a shared Wasserstein metric via Gaussian Wasserstein barycenters and uses geometry conflict to gate geometry-aware correction. Across Qwen3 0.6B--14B on domain-continual and capability-continual settings, GCWM consistently outperforms data-free baselines, improving retention and final performance without replay data. These results identify geometry conflict as both an explanatory signal for forgetting and a practical control signal for LLM continual post-training.

PiRD: Physics-informed Residual Diffusion for Flow Field Reconstruction

The use of machine learning in fluid dynamics is becoming more common to expedite the computation when solving forward and inverse problems of partial differential equations. Yet, a notable challenge with existing convolutional neural network (CNN)-based methods for data fidelity enhancement is their reliance on specific low-fidelity data patterns and distributions during the training phase. In addition, the CNN-based method essentially treats the flow reconstruction task as a computer vision task that prioritizes the element-wise precision which lacks a physical and mathematical explanation. This dependence can dramatically affect the models' effectiveness in real-world scenarios, especially when the low-fidelity input deviates from the training data or contains noise not accounted for during training. The introduction of diffusion models in this context shows promise for improving performance and generalizability. Unlike direct mapping from a specific low-fidelity to a high-fidelity distribution, diffusion models learn to transition from any low-fidelity distribution towards a high-fidelity one. Our proposed model - Physics-informed Residual Diffusion, demonstrates the capability to elevate the quality of data from both standard low-fidelity inputs, to low-fidelity inputs with injected Gaussian noise, and randomly collected samples. By integrating physics-based insights into the objective function, it further refines the accuracy and the fidelity of the inferred high-quality data. Experimental results have shown that our approach can effectively reconstruct high-quality outcomes for two-dimensional turbulent flows from a range of low-fidelity input conditions without requiring retraining.