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Intelligent systems to support collaborative learning rely on real-time behavioral data, including language, audio, and video. However, noisy data, such as word errors in speech recognition, audio static or background noise, and facial mistracking in video, often limit the utility of multimodal data. It is an open question of how we can build reliable multimodal models in the face of substantial data noise. In this paper, we investigate the impact of data noise on the recognition of confusion and conflict moments during collaborative programming sessions by 25 dyads of elementary school learners. We measure language errors with word error rate (WER), audio noise with speech-to-noise ratio (SNR), and video errors with frame-by-frame facial tracking accuracy. The results showed that the model’s accuracy for detecting confusion and conflict in the language modality decreased drastically from 0.84 to 0.73 when the WER exceeded 20%. Similarly, in the audio modality, the model’s accuracy decreased sharply from 0.79 to 0.61 when the SNR dropped below 5 dB. Conversely, the model’s accuracy remained relatively constant in the video modality at a comparable level (> 0.70) so long as at least one learner’s face was successfully tracked. Moreover, we trained several multimodal models and found that integrating multimodal data could effectively offset the negative effect of noise in unimodal data, ultimately leading to improved accuracy in recognizing confusion and conflict. These findings have practical implications for the future deployment of intelligent systems that support collaborative learning in actual classroom settings. 

The majority of computer algebra systems (CAS) support symbolic integration using a combination of heuristic algebraic and rule-based (integration table) methods. In this paper, we present a hybrid (symbolic-numeric) method to calculate the indefinite integrals of univariate expressions. Our method is broadly similar to the Risch-Norman algorithm. The primary motivation for this work is to add symbolic integration functionality to a modern CAS (the symbolic manipulation packages of SciML, the Scientific Machine Learning ecosystem of the Julia programming language), which is designed for numerical and machine learning applications. The symbolic part of our method is based on the combination of candidate terms generation (ansatz generation using a methodology borrowed from the Homotopy operators theory) combined with rule-based expression transformations provided by the underlying CAS. The numeric part uses sparse regression, a component of the Sparse Identification of Nonlinear Dynamics (SINDy) technique, to find the coefficients of the candidate terms. We show that this system can solve a large variety of common integration problems using only a few dozen basic integration rules.

 

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. Exploiting this feature, we demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We illustrate how this symbolic system improves numerical computing tasks by showcasing an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation. 

Modern design, control, and optimization often require multiple expensive simulations of highly nonlinear stiff models. These costs can be amortized by training a cheap surrogate of the full model, which can then be used repeatedly. Here we present a general data-driven method, the continuous time echo state network (CTESN), for generating surrogates of nonlinear ordinary differential equations with dynamics at widely separated timescales. We empirically demonstrate the ability to accelerate a physically motivated scalable model of a heating system by 98x while maintaining relative error of within 0.2 %. We showcase the ability for this surrogate to accurately handle highly stiff systems which have been shown to cause training failures with common surrogate methods such as Physics-Informed Neural Networks (PINNs), Long Short Term Memory (LSTM) networks, and discrete echo state networks (ESN). We show that our model captures fast transients as well as slow dynamics, while demonstrating that fixed time step machine learning techniques are unable to adequately capture the multi-rate behavior. Together this provides compelling evidence for the ability of CTESN surrogates to predict and accelerate highly stiff dynamical systems which are unable to be directly handled by previous scientific machine learning techniques.