We propose the first learning scheme for functional differential equations (FDEs). FDEs play a fundamental role in physics, mathematics, and optimal control.

Despite its widespread use in neural networks, error backpropagation has faced criticism for its lack of biological plausibility, suffering from issues such as the backward locking problem and the weight transport problem.

Such models, recently coined foundation models,h a v eb e e n transformational to the field of natural language processing.

Electromagnetic field simulation is central to designing, optimizing, and validating photonic devices and circuits. However, costly computation associated with numerical simulation poses a significant bottleneck, hindering scalability and turnaround time in the photonic circuit design process. Neural operators offer a promising alternative, but existing SOT A approaches, NeurOLight, struggle with predicting high-fidelity fields for real-world complicated photonic devices, with the best reported 0.38 normalized mean absolute error in NeurOLight. The interplays of highly complex light-matter interaction, e.g., scattering and resonance, sensitivity to local structure details, non-uniform learning complexity for full-domain simulation, and rich frequency information, contribute to the failure of existing neural PDE solvers. In this work, we boost the prediction fidelity to an unprecedented level for simulating complex photonic devices with a novel operator design driven by the above challenges. We propose a novel cross-axis fac-torized PACE operator with a strong long-distance modeling capacity to connect the full-domain complex field pattern with local device structures. Inspired by human learning, we further divide and conquer the simulation task for extremely hard cases into two progressively easy tasks, with a first-stage model learning an initial solution refined by a second model. On various complicated photonic device benchmarks, we demonstrate one sole PACE model is capable of achieving 73% lower error with 50% fewer parameters compared with various recent ML for PDE solvers.

Teaching a machine to "see, understand, and reason" the physical world like humans has been a

Figure 1: Comparison of JKO-iFlow (proposed) and standard CNF models.

Multi-fidelity fusion has become an important surrogate technique, which provides insights into expensive computer simulations and effectively improves decision-making, e.g., optimization, with less computational cost. Multi-fidelity fusion is much more computationally efficient compared to traditional single-fidelity surrogates. Despite the fast advancement of multi-fidelity fusion techniques, they lack a systematic framework to make use of the fidelity indicator, deal with high-dimensional and arbitrary data structure, and scale well to infinite-fidelity problems. In this work, we first generalize the popular autoregression (AR) to derive a novel linear fidelity differential equation (FiDE), paving the way to tractable infinite-fidelity fusion. We generalize FiDE to a high-dimensional system, which also provides a unifying framework to seemly bridge the gap between many multi-and single-fidelity GP-based models. We then propose ContinuAR, a rank-1 approximation solution to FiDEs, which is tractable to train, compatible with arbitrary multi-fidelity data structure, linearly scalable to the output dimension, and most importantly, delivers consistent SOT A performance with a significant margin over the baseline methods. Compared to the SOT A infinite-fidelity fusion, IFC, ContinuAR achieves up to 4x improvement in accuracy and 62,500x speedup in training time.

Google's latest Gemini AI upgrades attempt to anticipate what useful information you made need from your life to address a potential issue, make you to better photographer or become your personalised health and sleep coach. Shipping on the just-announced Pixel 10 Android phones, the new Magic Cue feature enables the chatbot to comb through your digital life and pull up relevant information on your phone just when you need it. Placing a call to an airline will automatically display your booking information from Gmail in the phone app. Or when a friend texts about brunch on Sunday Gemini will suggest a suitable coffee shop and show your calendar in line with your messages. The feature is part of a series of artificial intelligence upgrades for the newly announced Pixel 10, 10 Pro and 10 Pro Fold phones.

The proposed methodology can easily deal with large meshes without the need for explicit shape parameterization and provides crucial predictive uncertainties, which are essential for informed decision-making.