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

Predict+Optimize, a novel extension catering to applications where unknown parameters are instead revealed in sequential stages, with optimization decisions made in between. We further develop three training algorithms for neural networks (NNs) for our framework as proof of concept, all of which can handle mixed integer linear programs.

The dataset is obtained from Li et al. (2022a), which consists of an interpolated dataset of Fourier layers with mode 12 and width 32 are used. F-FNO: Following the settings in Li et al. (2022a), we train the F-FNO model (Li et al., UNet: Analogous to the setup in Li et al. (2022a), we train a UNet model (Ronneberger A typical training curve can be found in Figure 8. Table 4: The per-epoch runtime (in seconds) of selected models for the hyperelasticity problem. We note that the numbers of trainable parameters for the "Geo-FNO" and "FNO" cases are different from The airfoil dataset is directly taken from Li et al. (2022a), which is an interpolated dataset of The physical parameters used in generating the data are: Y oung's modulus Symmetry is enforced only when the topology characteristic function χ is updated. Besides the resolution-independence property of DAFNO as shown in Figure 3, we further investigate the generalizability of DAFNO in both physical and temporal resolutions with this example. Specifically, the eDAFNO model is trained on a spatial resolution of 64 64 and a time step of 0.02 Our results show that eDAFNO prediction remains independent of the time step employed.

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.

By incorporating the entropy term, we can ensure the diversity of the generated environments.

Inverse optimization has been increasingly used to estimate unknown parameters in an optimization model based on decision data.