Recent advances in operator learning have produced two distinct approaches for solving partial differential equations (PDEs): attention-based methods offering point-level adaptability but lacking spectral constraints, and spectral-based methods providing domain-level continuity priors but limited in local flexibility. This dichotomy has hindered the development of PDE solvers with both strong flexibility and generalization capability. This work introduces Holistic Physics Mixer (HPM), a novel framework that bridges this gap by integrating spectral and physical information in a unified space. HPM unifies both approaches as special cases while enabling more powerful spectral-physical interactions beyond either method alone. This enables HPM to inherit both the strong generalization of spectral methods and the flexibility of attention mechanisms while avoiding their respective limitations. Through extensive experiments across diverse PDE problems, we demonstrate that HPM consistently outperforms state-of-the-art methods in both accuracy and computational efficiency, while maintaining strong generalization capabilities with limited training data and excellent zero-shot performance on unseen resolutions.

The limited availability of high-quality training data poses a major obstacle in data-driven PDE solving, where expensive data collection and resolution constraints severely impact the ability of neural operator networks to learn and generalize the underlying physical system. To address this challenge, we propose DeltaPhi, a novel learning framework that transforms the PDE solving task from learning direct input-output mappings to learning the residuals between similar physical states, a fundamentally different approach to neural operator learning. This reformulation provides implicit data augmentation by exploiting the inherent stability of physical systems where closer initial states lead to closer evolution trajectories. DeltaPhi is architecture-agnostic and can be seamlessly integrated with existing neural operators to enhance their performance. Extensive experiments demonstrate consistent and significant improvements across diverse physical systems including regular and irregular domains, different neural architectures, multiple training data amount, and cross-resolution scenarios, confirming its effectiveness as a general enhancement for neural operators in data-limited PDE solving.

Audio-driven talking-head synthesis is a popular research topic for virtual human-related applications. However, the inflexibility and inefficiency of existing methods, which necessitate expensive end-to-end training to transfer emotions from guidance videos to talking-head predictions, are significant limitations. In this work, we propose the Emotional Adaptation for Audio-driven Talking-head (EAT) method, which transforms emotion-agnostic talking-head models into emotion-controllable ones in a cost-effective and efficient manner through parameter-efficient adaptations. Our approach utilizes a pretrained emotion-agnostic talking-head transformer and introduces three lightweight adaptations (the Deep Emotional Prompts, Emotional Deformation Network, and Emotional Adaptation Module) from different perspectives to enable precise and realistic emotion controls. Our experiments demonstrate that our approach achieves state-of-the-art performance on widely-used benchmarks, including LRW and MEAD. Additionally, our parameter-efficient adaptations exhibit remarkable generalization ability, even in scenarios where emotional training videos are scarce or nonexistent. Project website: https://yuangan.github.io/eat/