Josiah Bjorgaard

Anuj Kumar, Josiah Bjorgaard, Nikolaos Bouklas et al.

We introduce "Courant", a Perceiver-based encoder-processor-decoder surrogate model that has latent features exhibiting adaptive specialization and local support in the physical space, enabling functionality akin to an adaptive hp-refinement scheme, an attribute that is highly desirable in traditional numerical solvers and scientific machine learning broadly. The proposed architecture combines a shared random Fourier feature coordinate embedding, state-adapted latent queries, and a light-weight decoder. Courant is trained end-to-end with steady or transient simulation data and only a standard L_2 prediction loss in the physical space, achieving competitive accuracy on benchmarks. We demonstrate that Courant's inductive biases yield latents that are interpretable by design: they develop multiscale geometric specialization in the simulation domain and track coherent structures in the time-dependent case, acting analogously to time-evolving spatial basis functions and allowing for decoding a compact, geometry-anchored, partition-of-unity-like decomposition of the simulated field.

Nikolas Borrel-Jensen, Josiah Bjorgaard

We demonstrate a practical differentiable programming approach for acoustic inverse problems through two applications: admittance estimation and shape optimization for resonance damping. First, we show that JAX-FEM's automatic differentiation (AD) enables direct gradient-based estimation of complex boundary admittance from sparse pressure measurements, achieving 3-digit precision without requiring manual derivation of adjoint equations. Second, we apply randomized finite differences to acoustic shape optimization, combining JAX-FEM for forward simulation with PyTorch3D for mesh manipulation through AD. By separating physics-driven boundary optimization from geometry-driven interior mesh adaptation, we achieve 48.1% energy reduction at target frequencies with 30-fold fewer FEM solutions compared to standard finite difference on the full mesh. This work showcases how modern differentiable software stacks enable rapid prototyping of optimization workflows for physics-based inverse problems, with automatic differentiation for parameter estimation and a combination of finite differences and AD for geometric design.

Andrin Rehmann, Nolan Black, Josiah Bjorgaard et al.

Gradient-based optimization of engineering designs is limited by non-differentiable components in the typical computer-aided engineering (CAE) workflow, which calculates performance metrics from design parameters. While gradient-based methods could provide noticeable speed-ups in high-dimensional design spaces, codes for meshing, physical simulations, and other common components are not differentiable even if the math or physics underneath them is. We propose replacing non-differentiable pipeline components with surrogate models which are inherently differentiable. Using a toy example of aerodynamic shape optimization, we demonstrate an end-to-end differentiable pipeline where a 3D U-Net full-field surrogate replaces both meshing and simulation steps by training it on the mapping between the signed distance field (SDF) of the shape and the fields of interest. This approach enables gradient-based shape optimization without the need for differentiable solvers, which can be useful in situations where adjoint methods are unavailable and/or hard to implement.

Josiah Bjorgaard

Multimodal data fusion is essential for applications requiring the integration of diverse data sources, especially in the presence of incomplete or sparsely available modalities. This paper presents a comparative study of three multimodal embedding techniques, Modal Channel Attention (MCA), Zorro, and Everything at Once (EAO), to evaluate their performance on sparsely multimodal data. MCA introduces fusion embeddings for all combinations of input modalities and uses attention masking to create distinct attention channels, enabling flexible and efficient data fusion. Experiments on two datasets with four modalities each, CMU-MOSEI and TCGA, demonstrate that MCA outperforms Zorro across ranking, recall, regression, and classification tasks and outperforms EAO across regression and classification tasks. MCA achieves superior performance by maintaining robust uniformity across unimodal and fusion embeddings. While EAO performs best in ranking metrics due to its approach of forming fusion embeddings post-inference, it underperforms in downstream tasks requiring multimodal interactions. These results highlight the importance of contrasting all modality combinations in constructing embedding spaces and offers insights into the design of multimodal architectures for real-world applications with incomplete data.