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1. Neural Flow Map Reconstruction

   https://doi.org/10.1111/cgf.14549  

   Sahoo, S.; Lu, Y.; Berger, M. (August 2022, Computer Graphics Forum)

   Abstract In this paper we present a reconstruction technique for the reduction of unsteady flow data based on neural representations of time‐varying vector fields. Our approach is motivated by the large amount of data typically generated in numerical simulations, and in turn the types of data that domain scientists can generatein situthat are compact, yet useful, for post hoc analysis. One type of data commonly acquired during simulation are samples of the flow map, where a single sample is the result of integrating the underlying vector field for a specified time duration. In our work, we treat a collection of flow map samples for a single dataset as a meaningful, compact, and yet incomplete, representation of unsteady flow, and our central objective is to find a representation that enables us to best recover arbitrary flow map samples. To this end, we introduce a technique for learning implicit neural representations of time‐varying vector fields that are specifically optimized to reproduce flow map samples sparsely covering the spatiotemporal domain of the data. We show that, despite aggressive data reduction, our optimization problem — learning a function‐space neural network to reproduce flow map samples under a fixed integration scheme — leads to representations that demonstrate strong generalization, both in the field itself, and using the field to approximate the flow map. Through quantitative and qualitative analysis across different datasets we show that our approach is an improvement across a variety of data reduction methods, and across a variety of measures ranging from improved vector fields, flow maps, and features derived from the flow map. 

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2. SLANG.D: Fast, Modular and Differentiable Shader Programming

   https://doi.org/10.1145/3618353  

   Bangaru, Sai_Praveen; Wu, Lifan; Li, Tzu-Mao; Munkberg, Jacob; Bernstein, Gilbert; Ragan-Kelley, Jonathan; Durand, Frédo; Lefohn, Aaron; He, Yong (December 2023, ACM Transactions on Graphics)

   We introduce SLANG.D, an extension to the Slang shading language that incorporates first-class automatic differentiation support. The new shading language allows us to transform a Direct3D-based path tracer to be fully differentiable with minor modifications to existing code. SLANG.D enables a shared ecosystem between machine learning frameworks and pre-existing graphics hardware API-based rendering systems, promoting the interchange of components and ideas across these two domains. Our contributions include a differentiable type system designed to ensure type safety and semantic clarity in codebases that blend differentiable and non-differentiable code, language primitives that automatically generate both forward and reverse gradient propagation methods, and a compiler architecture that generates efficient derivative propagation shader code for graphics pipelines. Our compiler supports differentiating code that involves arbitrary control-flow, dynamic dispatch, generics and higher-order differentiation, while providing developers flexible control of checkpointing and gradient aggregation strategies for best performance. Our system allows us to differentiate an existing real-time path tracer, Falcor, with minimal change to its shader code. We show that the compiler-generated derivative kernels perform as efficiently as handwritten ones. In several benchmarks, the SLANG.D code achieves significant speedup when compared to prior automatic differentiation systems. 

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3. On infinitesimal contraction analysis for hybrid systems

   Burden, S; Coogan, S (December 2022, IEEE Conference on Decision and Control)

   Infinitesimal contraction analysis, wherein global convergence results are obtained from properties of local dynamics, is a powerful analysis tool. In this paper, we generalize infinitesimal contraction analysis to hybrid systems in which state-dependent guards trigger transitions defined by reset maps between modes that may have different norms and need not be of the same dimension. In contrast to existing literature, we do not restrict mode sequence or dwell time. We work in settings where the hybrid system flow is differentiable almost everywhere and its derivative is the solution to a jump-linear-time-varying differential equation whose jumps are defined by a saltation matrix determined from the guard, reset map, and vector field. Our main result shows that if the vector field is infinitesimally contracting, and if the saltation matrix is non-expansive, then the intrinsic distance between any two trajectories decreases exponentially in time. When bounds on dwell time are available, our approach yields a bound on the intrinsic distance between trajectories regardless of whether the dynamics are expansive or contractive. We illustrate our results using wo examples: a constrained mechanical system and an electrical circuit with an ideal diode. 

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4. On infinitesimal contraction analysis for hybrid systems

   Burden, Samuel A.; Coogan, Samuel D. (December 2022, IEEE Conference on Decision and Control)

   Infinitesimal contraction analysis, wherein global convergence results are obtained from properties of local dynamics, is a powerful analysis tool. In this letter, we generalize infinitesimal contraction analysis to hybrid systems in which state-dependent guards trigger transitions defined by reset maps between modes that may have different norms and need not be of the same dimension. In contrast to existing literature, we do not restrict mode sequence or dwell time. We work in settings where the hybrid system flow is differentiable almost everywhere and its derivative is the solution to a jump-linear-time-varying differential equation whose jumps are defined by a saltation matrix determined from the guard, reset map, and vector field. Our main result shows that if the vector field is infinitesimally contracting, and if the saltation matrix is non-expansive, then the intrinsic distance between any two trajectories decreases exponentially in time. When bounds on dwell time are available, our approach yields a bound on the intrinsic distance between trajectories regardless of whether the dynamics are expansive or contractive. We illustrate our results using two examples: a constrained mechanical system and an electrical circuit with an ideal diode. 

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5. Regularity of the Level Set Flow

   https://doi.org/10.1002/cpa.21703  

   Colding, Tobias Holck; Minicozzi, William P (July 2017, Communications on Pure and Applied Mathematics)

   Abstract We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative and satisfies the equation classically. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closedC¹manifold with cylindrical singularities. © 2017 Wiley Periodicals, Inc. 

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