Michael Poli

Eric Nguyen, Michael Poli, Marjan Faizi et al.

Genomic (DNA) sequences encode an enormous amount of information for gene regulation and protein synthesis. Similar to natural language models, researchers have proposed foundation models in genomics to learn generalizable features from unlabeled genome data that can then be fine-tuned for downstream tasks such as identifying regulatory elements. Due to the quadratic scaling of attention, previous Transformer-based genomic models have used 512 to 4k tokens as context (<0.001% of the human genome), significantly limiting the modeling of long-range interactions in DNA. In addition, these methods rely on tokenizers or fixed k-mers to aggregate meaningful DNA units, losing single nucleotide resolution where subtle genetic variations can completely alter protein function via single nucleotide polymorphisms (SNPs). Recently, Hyena, a large language model based on implicit convolutions was shown to match attention in quality while allowing longer context lengths and lower time complexity. Leveraging Hyena's new long-range capabilities, we present HyenaDNA, a genomic foundation model pretrained on the human reference genome with context lengths of up to 1 million tokens at the single nucleotide-level - an up to 500x increase over previous dense attention-based models. HyenaDNA scales sub-quadratically in sequence length (training up to 160x faster than Transformer), uses single nucleotide tokens, and has full global context at each layer. We explore what longer context enables - including the first use of in-context learning in genomics. On fine-tuned benchmarks from the Nucleotide Transformer, HyenaDNA reaches state-of-the-art (SotA) on 12 of 18 datasets using a model with orders of magnitude less parameters and pretraining data. On the GenomicBenchmarks, HyenaDNA surpasses SotA on 7 of 8 datasets on average by +10 accuracy points. Code at https://github.com/HazyResearch/hyena-dna.

Michael Zhang, Khaled K. Saab, Michael Poli et al. · stanford

Time series modeling is a well-established problem, which often requires that methods (1) expressively represent complicated dependencies, (2) forecast long horizons, and (3) efficiently train over long sequences. State-space models (SSMs) are classical models for time series, and prior works combine SSMs with deep learning layers for efficient sequence modeling. However, we find fundamental limitations with these prior approaches, proving their SSM representations cannot express autoregressive time series processes. We thus introduce SpaceTime, a new state-space time series architecture that improves all three criteria. For expressivity, we propose a new SSM parameterization based on the companion matrix -- a canonical representation for discrete-time processes -- which enables SpaceTime's SSM layers to learn desirable autoregressive processes. For long horizon forecasting, we introduce a "closed-loop" variation of the companion SSM, which enables SpaceTime to predict many future time-steps by generating its own layer-wise inputs. For efficient training and inference, we introduce an algorithm that reduces the memory and compute of a forward pass with the companion matrix. With sequence length $\ell$ and state-space size $d$, we go from $\tilde{O}(d \ell)$ naïvely to $\tilde{O}(d + \ell)$. In experiments, our contributions lead to state-of-the-art results on extensive and diverse benchmarks, with best or second-best AUROC on 6 / 7 ECG and speech time series classification, and best MSE on 14 / 16 Informer forecasting tasks. Furthermore, we find SpaceTime (1) fits AR($p$) processes that prior deep SSMs fail on, (2) forecasts notably more accurately on longer horizons than prior state-of-the-art, and (3) speeds up training on real-world ETTh1 data by 73% and 80% relative wall-clock time over Transformers and LSTMs.

Michael Poli, Stefano Massaroli, Eric Nguyen et al.

Recent advances in deep learning have relied heavily on the use of large Transformers due to their ability to learn at scale. However, the core building block of Transformers, the attention operator, exhibits quadratic cost in sequence length, limiting the amount of context accessible. Existing subquadratic methods based on low-rank and sparse approximations need to be combined with dense attention layers to match Transformers, indicating a gap in capability. In this work, we propose Hyena, a subquadratic drop-in replacement for attention constructed by interleaving implicitly parametrized long convolutions and data-controlled gating. In recall and reasoning tasks on sequences of thousands to hundreds of thousands of tokens, Hyena improves accuracy by more than 50 points over operators relying on state-spaces and other implicit and explicit methods, matching attention-based models. We set a new state-of-the-art for dense-attention-free architectures on language modeling in standard datasets (WikiText103 and The Pile), reaching Transformer quality with a 20% reduction in training compute required at sequence length 2K. Hyena operators are twice as fast as highly optimized attention at sequence length 8K, and 100x faster at sequence length 64K.

Keshigeyan Chandrasegaran, Kyle Sargent, Suchir Agarwal et al.

Studying scalable methods for visual generative modeling requires large, accessible, and stable datasets. We introduce GPIC, a Giant Permissive Image Corpus of approximately 28 trillion pixels. GPIC comprises diverse internet images captioned by a state-of-the-art vision-language model, including 100M training, 200K validation, and 1M test examples. Moreover, all GPIC images are permissively licensed for both research and commercial use. GPIC is safety-filtered, deduplicated, and centrally hosted on Hugging Face. We provide a benchmarking protocol for generative modeling on GPIC. Finally, we provide a reference baseline for pixel-space flow matching on GPIC. Our dataset, benchmark, and models are available at https://huggingface.co/datasets/stanford-vision-lab/gpic. Evaluation toolkit and code are available at https://gpic.stanford.edu

Linqi Zhou, Michael Poli, Winnie Xu et al.

Methods based on ordinary differential equations (ODEs) are widely used to build generative models of time-series. In addition to high computational overhead due to explicitly computing hidden states recurrence, existing ODE-based models fall short in learning sequence data with sharp transitions - common in many real-world systems - due to numerical challenges during optimization. In this work, we propose LS4, a generative model for sequences with latent variables evolving according to a state space ODE to increase modeling capacity. Inspired by recent deep state space models (S4), we achieve speedups by leveraging a convolutional representation of LS4 which bypasses the explicit evaluation of hidden states. We show that LS4 significantly outperforms previous continuous-time generative models in terms of marginal distribution, classification, and prediction scores on real-world datasets in the Monash Forecasting Repository, and is capable of modeling highly stochastic data with sharp temporal transitions. LS4 sets state-of-the-art for continuous-time latent generative models, with significant improvement of mean squared error and tighter variational lower bounds on irregularly-sampled datasets, while also being x100 faster than other baselines on long sequences.

Tri Dao, Beidi Chen, Nimit Sohoni et al.

Large neural networks excel in many domains, but they are expensive to train and fine-tune. A popular approach to reduce their compute or memory requirements is to replace dense weight matrices with structured ones (e.g., sparse, low-rank, Fourier transform). These methods have not seen widespread adoption (1) in end-to-end training due to unfavorable efficiency--quality tradeoffs, and (2) in dense-to-sparse fine-tuning due to lack of tractable algorithms to approximate a given dense weight matrix. To address these issues, we propose a class of matrices (Monarch) that is hardware-efficient (they are parameterized as products of two block-diagonal matrices for better hardware utilization) and expressive (they can represent many commonly used transforms). Surprisingly, the problem of approximating a dense weight matrix with a Monarch matrix, though nonconvex, has an analytical optimal solution. These properties of Monarch matrices unlock new ways to train and fine-tune sparse and dense models. We empirically validate that Monarch can achieve favorable accuracy-efficiency tradeoffs in several end-to-end sparse training applications: speeding up ViT and GPT-2 training on ImageNet classification and Wikitext-103 language modeling by 2x with comparable model quality, and reducing the error on PDE solving and MRI reconstruction tasks by 40%. In sparse-to-dense training, with a simple technique called "reverse sparsification," Monarch matrices serve as a useful intermediate representation to speed up GPT-2 pretraining on OpenWebText by 2x without quality drop. The same technique brings 23% faster BERT pretraining than even the very optimized implementation from Nvidia that set the MLPerf 1.1 record. In dense-to-sparse fine-tuning, as a proof-of-concept, our Monarch approximation algorithm speeds up BERT fine-tuning on GLUE by 1.7x with comparable accuracy.

Michael Poli, Stefano Massaroli, Federico Berto et al.

Spectral analysis provides one of the most effective paradigms for information-preserving dimensionality reduction, as simple descriptions of naturally occurring signals are often obtained via few terms of periodic basis functions. In this work, we study deep neural networks designed to harness the structure in frequency domain for efficient learning of long-range correlations in space or time: frequency-domain models (FDMs). Existing FDMs are based on complex-valued transforms i.e. Fourier Transforms (FT), and layers that perform computation on the spectrum and input data separately. This design introduces considerable computational overhead: for each layer, a forward and inverse FT. Instead, this work introduces a blueprint for frequency domain learning through a single transform: transform once (T1). To enable efficient, direct learning in the frequency domain we derive a variance-preserving weight initialization scheme and investigate methods for frequency selection in reduced-order FDMs. Our results noticeably streamline the design process of FDMs, pruning redundant transforms, and leading to speedups of 3x to 10x that increase with data resolution and model size. We perform extensive experiments on learning the solution operator of spatio-temporal dynamics, including incompressible Navier-Stokes, turbulent flows around airfoils and high-resolution video of smoke. T1 models improve on the test performance of FDMs while requiring significantly less computation (5 hours instead of 32 for our large-scale experiment), with over 20% reduction in average predictive error across tasks.

Stefano Massaroli, Michael Poli, Daniel Y. Fu et al.

Recent advances in attention-free sequence models rely on convolutions as alternatives to the attention operator at the core of Transformers. In particular, long convolution sequence models have achieved state-of-the-art performance in many domains, but incur a significant cost during auto-regressive inference workloads -- naively requiring a full pass (or caching of activations) over the input sequence for each generated token -- similarly to attention-based models. In this paper, we seek to enable $\mathcal O(1)$ compute and memory cost per token in any pre-trained long convolution architecture to reduce memory footprint and increase throughput during generation. Concretely, our methods consist in extracting low-dimensional linear state-space models from each convolution layer, building upon rational interpolation and model-order reduction techniques. We further introduce architectural improvements to convolution-based layers such as Hyena: by weight-tying the filters across channels into heads, we achieve higher pre-training quality and reduce the number of filters to be distilled. The resulting model achieves 10x higher throughput than Transformers and 1.5x higher than Hyena at 1.3B parameters, without any loss in quality after distillation.

Daniel Y. Fu, Simran Arora, Jessica Grogan et al.

Machine learning models are increasingly being scaled in both sequence length and model dimension to reach longer contexts and better performance. However, existing architectures such as Transformers scale quadratically along both these axes. We ask: are there performant architectures that can scale sub-quadratically along sequence length and model dimension? We introduce Monarch Mixer (M2), a new architecture that uses the same sub-quadratic primitive along both sequence length and model dimension: Monarch matrices, a simple class of expressive structured matrices that captures many linear transforms, achieves high hardware efficiency on GPUs, and scales sub-quadratically. As a proof of concept, we explore the performance of M2 in three domains: non-causal BERT-style language modeling, ViT-style image classification, and causal GPT-style language modeling. For non-causal BERT-style modeling, M2 matches BERT-base and BERT-large in downstream GLUE quality with up to 27% fewer parameters, and achieves up to 9.1$\times$ higher throughput at sequence length 4K. On ImageNet, M2 outperforms ViT-b by 1% in accuracy, with only half the parameters. Causal GPT-style models introduce a technical challenge: enforcing causality via masking introduces a quadratic bottleneck. To alleviate this bottleneck, we develop a novel theoretical view of Monarch matrices based on multivariate polynomial evaluation and interpolation, which lets us parameterize M2 to be causal while remaining sub-quadratic. Using this parameterization, M2 matches GPT-style Transformers at 360M parameters in pretraining perplexity on The PILE--showing for the first time that it may be possible to match Transformer quality without attention or MLPs.

Michael Poli, Winnie Xu, Stefano Massaroli et al.

Many patterns in nature exhibit self-similarity: they can be compactly described via self-referential transformations. Said patterns commonly appear in natural and artificial objects, such as molecules, shorelines, galaxies and even images. In this work, we investigate the role of learning in the automated discovery of self-similarity and in its utilization for downstream tasks. To this end, we design a novel class of implicit operators, Neural Collages, which (1) represent data as the parameters of a self-referential, structured transformation, and (2) employ hypernetworks to amortize the cost of finding these parameters to a single forward pass. We investigate how to leverage the representations produced by Neural Collages in various tasks, including data compression and generation. Neural Collages image compressors are orders of magnitude faster than other self-similarity-based algorithms during encoding and offer compression rates competitive with implicit methods. Finally, we showcase applications of Neural Collages for fractal art and as deep generative models.

Michael Poli, Stefano Massaroli, Stefano Ermon et al.

We present a methodology for formulating simplifying abstractions in machine learning systems by identifying and harnessing the utility structure of decisions. Machine learning tasks commonly involve high-dimensional output spaces (e.g., predictions for every pixel in an image or node in a graph), even though a coarser output would often suffice for downstream decision-making (e.g., regions of an image instead of pixels). Developers often hand-engineer abstractions of the output space, but numerous abstractions are possible and it is unclear how the choice of output space for a model impacts its usefulness in downstream decision-making. We propose a method that configures the output space automatically in order to minimize the loss of decision-relevant information. Taking a geometric perspective, we formulate a step of the algorithm as a projection of the probability simplex, termed fold, that minimizes the total loss of decision-related information in the H-entropy sense. Crucially, learning in the abstracted outcome space requires less data, leading to a net improvement in decision quality. We demonstrate the method in two domains: data acquisition for deep neural network training and a closed-loop wildfire management task.

Federico Berto, Stefano Massaroli, Michael Poli et al.

Synthesizing optimal controllers for dynamical systems often involves solving optimization problems with hard real-time constraints. These constraints determine the class of numerical methods that can be applied: computationally expensive but accurate numerical routines are replaced by fast and inaccurate methods, trading inference time for solution accuracy. This paper provides techniques to improve the quality of optimized control policies given a fixed computational budget. We achieve the above via a hypersolvers approach, which hybridizes a differential equation solver and a neural network. The performance is evaluated in direct and receding-horizon optimal control tasks in both low and high dimensions, where the proposed approach shows consistent Pareto improvements in solution accuracy and control performance.

Dragos Secrieru, Garyk Brixi, Yoshua Bengio et al.

Multi-hybrid architectures are poised to take over language modeling due to better quality and performance. We introduce a hierarchical decomposition framework for linear recurrences that allows us to develop algorithms aligned with GPU memory hierarchies, yielding Sliding Window Recurrences. We focus specifically on truncating recurrences to hardware-aligned windows which are naturally jagged, limiting costly inter-warp communication. Using SWR, we develop Phalanx layers that serve as drop-in replacements for windowed attention or linear recurrences. In 1B parameter multi-hybrid models, Phalanx achieves over 10-40% speedup across 4K to 32K context length over optimized Transformers while matching perplexity.

Chuanbo Hua, Federico Berto, Michael Poli et al.

While complex simulations of physical systems have been widely used in engineering and scientific computing, lowering their often prohibitive computational requirements has only recently been tackled by deep learning approaches. In this paper, we present GraphSplineNets, a novel deep-learning method to speed up the forecasting of physical systems by reducing the grid size and number of iteration steps of deep surrogate models. Our method uses two differentiable orthogonal spline collocation methods to efficiently predict response at any location in time and space. Additionally, we introduce an adaptive collocation strategy in space to prioritize sampling from the most important regions. GraphSplineNets improve the accuracy-speedup tradeoff in forecasting various dynamical systems with increasing complexity, including the heat equation, damped wave propagation, Navier-Stokes equations, and real-world ocean currents in both regular and irregular domains.

Simran Arora, Sabri Eyuboglu, Aman Timalsina et al.

Attention-free language models that combine gating and convolutions are growing in popularity due to their efficiency and increasingly competitive performance. To better understand these architectures, we pretrain a suite of 17 attention and "gated-convolution" language models, finding that SoTA gated-convolution architectures still underperform attention by up to 2.1 perplexity points on the Pile. In fine-grained analysis, we find 82% of the gap is explained by each model's ability to recall information that is previously mentioned in-context, e.g. "Hakuna Matata means no worries Hakuna Matata it means no" $\rightarrow$ "??". On this task, termed "associative recall", we find that attention outperforms gated-convolutions by a large margin: a 70M parameter attention model outperforms a 1.4 billion parameter gated-convolution model on associative recall. This is surprising because prior work shows gated convolutions can perfectly solve synthetic tests for AR capability. To close the gap between synthetics and real language, we develop a new formalization of the task called multi-query associative recall (MQAR) that better reflects actual language. We perform an empirical and theoretical study of MQAR that elucidates differences in the parameter-efficiency of attention and gated-convolution recall. Informed by our analysis, we evaluate simple convolution-attention hybrids and show that hybrids with input-dependent sparse attention patterns can close 97.4% of the gap to attention, while maintaining sub-quadratic scaling. Our code is accessible at: https://github.com/HazyResearch/zoology.

Rom N. Parnichkun, Stefano Massaroli, Alessandro Moro et al.

We approach designing a state-space model for deep learning applications through its dual representation, the transfer function, and uncover a highly efficient sequence parallel inference algorithm that is state-free: unlike other proposed algorithms, state-free inference does not incur any significant memory or computational cost with an increase in state size. We achieve this using properties of the proposed frequency domain transfer function parametrization, which enables direct computation of its corresponding convolutional kernel's spectrum via a single Fast Fourier Transform. Our experimental results across multiple sequence lengths and state sizes illustrates, on average, a 35% training speed improvement over S4 layers -- parametrized in time-domain -- on the Long Range Arena benchmark, while delivering state-of-the-art downstream performances over other attention-free approaches. Moreover, we report improved perplexity in language modeling over a long convolutional Hyena baseline, by simply introducing our transfer function parametrization. Our code is available at https://github.com/ruke1ire/RTF.

Michael Poli, Armin W Thomas, Eric Nguyen et al.

The development of deep learning architectures is a resource-demanding process, due to a vast design space, long prototyping times, and high compute costs associated with at-scale model training and evaluation. We set out to simplify this process by grounding it in an end-to-end mechanistic architecture design (MAD) pipeline, encompassing small-scale capability unit tests predictive of scaling laws. Through a suite of synthetic token manipulation tasks such as compression and recall, designed to probe capabilities, we identify and test new hybrid architectures constructed from a variety of computational primitives. We experimentally validate the resulting architectures via an extensive compute-optimal and a new state-optimal scaling law analysis, training over 500 language models between 70M to 7B parameters. Surprisingly, we find MAD synthetics to correlate with compute-optimal perplexity, enabling accurate evaluation of new architectures via isolated proxy tasks. The new architectures found via MAD, based on simple ideas such as hybridization and sparsity, outperform state-of-the-art Transformer, convolutional, and recurrent architectures (Transformer++, Hyena, Mamba) in scaling, both at compute-optimal budgets and in overtrained regimes. Overall, these results provide evidence that performance on curated synthetic tasks can be predictive of scaling laws, and that an optimal architecture should leverage specialized layers via a hybrid topology.

Jerome Ku, Eric Nguyen, David W. Romero et al.

We introduce convolutional multi-hybrid architectures, with a design grounded on two simple observations. First, operators in hybrid models can be tailored to token manipulation tasks such as in-context recall, multi-token recall, and compression, with input-dependent convolutions and attention offering complementary performance. Second, co-designing convolution operators and hardware-aware algorithms enables efficiency gains in regimes where previous alternative architectures struggle to surpass Transformers. At the 40 billion parameter scale, we train end-to-end 1.2 to 2.9 times faster than optimized Transformers, and 1.1 to 1.4 times faster than previous generation hybrids. On H100 GPUs and model width 4096, individual operators in the proposed multi-hybrid StripedHyena 2 architecture achieve two-fold throughput improvement over linear attention and state-space models. Multi-hybrids excel at sequence modeling over byte-tokenized data, as demonstrated by the Evo 2 line of models. We discuss the foundations that enable these results, including architecture design, overlap-add blocked kernels for tensor cores, and dedicated all-to-all and point-to-point context parallelism strategies.

Armin W. Thomas, Rom Parnichkun, Alexander Amini et al.

Iterative improvement of model architectures is fundamental to deep learning: Transformers first enabled scaling, and recent advances in model hybridization have pushed the quality-efficiency frontier. However, optimizing architectures remains challenging and expensive. Current automated or manual approaches fall short, largely due to limited progress in the design of search spaces and due to the simplicity of resulting patterns and heuristics. In this work, we propose a new approach for the synthesis of tailored architectures (STAR). Our approach combines a novel search space based on the theory of linear input-varying systems, supporting a hierarchical numerical encoding into architecture genomes. STAR genomes are automatically refined and recombined with gradient-free, evolutionary algorithms to optimize for multiple model quality and efficiency metrics. Using STAR, we optimize large populations of new architectures, leveraging diverse computational units and interconnection patterns, improving over highly-optimized Transformers and striped hybrid models on the frontier of quality, parameter size, and inference cache for autoregressive language modeling.

Keshigeyan Chandrasegaran, Michael Poli, Daniel Y. Fu et al. · salesforce, stanford

Designing model architectures requires decisions such as selecting operators (e.g., attention, convolution) and configurations (e.g., depth, width). However, evaluating the impact of these decisions on model quality requires costly pretraining, limiting architectural investigation. Inspired by how new software is built on existing code, we ask: can new architecture designs be studied using pretrained models? To this end, we present grafting, a simple approach for editing pretrained diffusion transformers (DiTs) to materialize new architectures under small compute budgets. Informed by our analysis of activation behavior and attention locality, we construct a testbed based on the DiT-XL/2 design to study the impact of grafting on model quality. Using this testbed, we develop a family of hybrid designs via grafting: replacing softmax attention with gated convolution, local attention, and linear attention, and replacing MLPs with variable expansion ratio and convolutional variants. Notably, many hybrid designs achieve good quality (FID: 2.38-2.64 vs. 2.27 for DiT-XL/2) using <2% pretraining compute. We then graft a text-to-image model (PixArt-Sigma), achieving a 1.43x speedup with less than a 2% drop in GenEval score. Finally, we present a case study that restructures DiT-XL/2 by converting every pair of sequential transformer blocks into parallel blocks via grafting. This reduces model depth by 2x and yields better quality (FID: 2.77) than other models of comparable depth. Together, we show that new diffusion model designs can be explored by grafting pretrained DiTs, with edits ranging from operator replacement to architecture restructuring. Code and grafted models: https://grafting.stanford.edu

André Hottung, Federico Berto, Chuanbo Hua et al. · pku

Designing high-performing heuristics for vehicle routing problems (VRPs) is a complex task that requires both intuition and deep domain knowledge. Large language model (LLM)-based code generation has recently shown promise across many domains, but it still falls short of producing heuristics that rival those crafted by human experts. In this paper, we propose VRPAgent, a framework that integrates LLM-generated components into a metaheuristic and refines them through a novel genetic search. By using the LLM to generate problem-specific operators, embedded within a generic metaheuristic framework, VRPAgent keeps tasks manageable, guarantees correctness, and still enables the discovery of novel and powerful strategies. Across multiple problems, including the capacitated VRP, the VRP with time windows, and the prize-collecting VRP, our method discovers heuristic operators that outperform handcrafted methods and recent learning-based approaches while requiring only a single CPU core. To our knowledge, \VRPAgent is the first LLM-based paradigm to advance the state-of-the-art in VRPs, highlighting a promising future for automated heuristics discovery.

Rom N. Parnichkun, Neehal Tumma, Armin W. Thomas et al.

The need to develop a general framework for architecture analysis is becoming increasingly important, given the expanding design space of sequence models. To this end, we draw insights from classical signal processing and control theory, to develop a quantitative measure of \textit{memory utilization}: the internal mechanisms through which a model stores past information to produce future outputs. This metric, which we call \textbf{\textit{effective state-size}} (ESS), is tailored to the fundamental class of systems with \textit{input-invariant} and \textit{input-varying linear operators}, encompassing a variety of computational units such as variants of attention, convolutions, and recurrences. Unlike prior work on memory utilization, which either relies on raw operator visualizations (e.g. attention maps), or simply the total \textit{memory capacity} (i.e. cache size) of a model, our metrics provide highly interpretable and actionable measurements. In particular, we show how ESS can be leveraged to improve initialization strategies, inform novel regularizers and advance the performance-efficiency frontier through model distillation. Furthermore, we demonstrate that the effect of context delimiters (such as end-of-speech tokens) on ESS highlights cross-architectural differences in how large language models utilize their available memory to recall information. Overall, we find that ESS provides valuable insights into the dynamics that dictate memory utilization, enabling the design of more efficient and effective sequence models.

Luca Scimeca, Seong Joon Oh, Sanghyuk Chun et al.

Deep neural networks (DNNs) often rely on easy-to-learn discriminatory features, or cues, that are not necessarily essential to the problem at hand. For example, ducks in an image may be recognized based on their typical background scenery, such as lakes or streams. This phenomenon, also known as shortcut learning, is emerging as a key limitation of the current generation of machine learning models. In this work, we introduce a set of experiments to deepen our understanding of shortcut learning and its implications. We design a training setup with several shortcut cues, named WCST-ML, where each cue is equally conducive to the visual recognition problem at hand. Even under equal opportunities, we observe that (1) certain cues are preferred to others, (2) solutions biased to the easy-to-learn cues tend to converge to relatively flat minima on the loss surface, and (3) the solutions focusing on those preferred cues are far more abundant in the parameter space. We explain the abundance of certain cues via their Kolmogorov (descriptional) complexity: solutions corresponding to Kolmogorov-simple cues are abundant in the parameter space and are thus preferred by DNNs. Our studies are based on the synthetic dataset DSprites and the face dataset UTKFace. In our WCST-ML, we observe that the inborn bias of models leans toward simple cues, such as color and ethnicity. Our findings emphasize the importance of active human intervention to remove the inborn model biases that may cause negative societal impacts.

Michael Poli, Stefano Massaroli, Clayton M. Rabideau et al.

We introduce the framework of continuous-depth graph neural networks (GNNs). Neural graph differential equations (Neural GDEs) are formalized as the counterpart to GNNs where the input-output relationship is determined by a continuum of GNN layers, blending discrete topological structures and differential equations. The proposed framework is shown to be compatible with static GNN models and is extended to dynamic and stochastic settings through hybrid dynamical system theory. Here, Neural GDEs improve performance by exploiting the underlying dynamics geometry, further introducing the ability to accommodate irregularly sampled data. Results prove the effectiveness of the proposed models across applications, such as traffic forecasting or prediction in genetic regulatory networks.

Michael Poli, Stefano Massaroli, Luca Scimeca et al.

Effective control and prediction of dynamical systems often require appropriate handling of continuous-time and discrete, event-triggered processes. Stochastic hybrid systems (SHSs), common across engineering domains, provide a formalism for dynamical systems subject to discrete, possibly stochastic, state jumps and multi-modal continuous-time flows. Despite the versatility and importance of SHSs across applications, a general procedure for the explicit learning of both discrete events and multi-mode continuous dynamics remains an open problem. This work introduces Neural Hybrid Automata (NHAs), a recipe for learning SHS dynamics without a priori knowledge on the number of modes and inter-modal transition dynamics. NHAs provide a systematic inference method based on normalizing flows, neural differential equations and self-supervision. We showcase NHAs on several tasks, including mode recovery and flow learning in systems with stochastic transitions, and end-to-end learning of hierarchical robot controllers.

Stefano Massaroli, Michael Poli, Sho Sonoda et al.

We detail a novel class of implicit neural models. Leveraging time-parallel methods for differential equations, Multiple Shooting Layers (MSLs) seek solutions of initial value problems via parallelizable root-finding algorithms. MSLs broadly serve as drop-in replacements for neural ordinary differential equations (Neural ODEs) with improved efficiency in number of function evaluations (NFEs) and wall-clock inference time. We develop the algorithmic framework of MSLs, analyzing the different choices of solution methods from a theoretical and computational perspective. MSLs are showcased in long horizon optimal control of ODEs and PDEs and as latent models for sequence generation. Finally, we investigate the speedups obtained through application of MSL inference in neural controlled differential equations (Neural CDEs) for time series classification of medical data.

Stefano Massaroli, Michael Poli, Stefano Peluchetti et al.

We systematically develop a learning-based treatment of stochastic optimal control (SOC), relying on direct optimization of parametric control policies. We propose a derivation of adjoint sensitivity results for stochastic differential equations through direct application of variational calculus. Then, given an objective function for a predetermined task specifying the desiderata for the controller, we optimize their parameters via iterative gradient descent methods. In doing so, we extend the range of applicability of classical SOC techniques, often requiring strict assumptions on the functional form of system and control. We verify the performance of the proposed approach on a continuous-time, finite horizon portfolio optimization with proportional transaction costs.

Stefano Massaroli, Michael Poli, Federico Califano et al.

We introduce optimal energy shaping as an enhancement of classical passivity-based control methods. A promising feature of passivity theory, alongside stability, has traditionally been claimed to be intuitive performance tuning along the execution of a given task. However, a systematic approach to adjust performance within a passive control framework has yet to be developed, as each method relies on few and problem-specific practical insights. Here, we cast the classic energy-shaping control design process in an optimal control framework; once a task-dependent performance metric is defined, an optimal solution is systematically obtained through an iterative procedure relying on neural networks and gradient-based optimization. The proposed method is validated on state-regulation tasks.

Daehoon Gwak, Gyuhyeon Sim, Michael Poli et al.

By interpreting the forward dynamics of the latent representation of neural networks as an ordinary differential equation, Neural Ordinary Differential Equation (Neural ODE) emerged as an effective framework for modeling a system dynamics in the continuous time domain. However, real-world systems often involves external interventions that cause changes in the system dynamics such as a moving ball coming in contact with another ball, or such as a patient being administered with particular drug. Neural ODE and a number of its recent variants, however, are not suitable for modeling such interventions as they do not properly model the observations and the interventions separately. In this paper, we propose a novel neural ODE-based approach (IMODE) that properly model the effect of external interventions by employing two ODE functions to separately handle the observations and the interventions. Using both synthetic and real-world time-series datasets involving interventions, our experimental results consistently demonstrate the superiority of IMODE compared to existing approaches.

Michael Poli, Stefano Massaroli, Atsushi Yamashita et al.

Continuous-depth learning has recently emerged as a novel perspective on deep learning, improving performance in tasks related to dynamical systems and density estimation. Core to these approaches is the neural differential equation, whose forward passes are the solutions of an initial value problem parametrized by a neural network. Unlocking the full potential of continuous-depth models requires a different set of software tools, due to peculiar differences compared to standard discrete neural networks, e.g inference must be carried out via numerical solvers. We introduce TorchDyn, a PyTorch library dedicated to continuous-depth learning, designed to elevate neural differential equations to be as accessible as regular plug-and-play deep learning primitives. This objective is achieved by identifying and subdividing different variants into common essential components, which can be combined and freely repurposed to obtain complex compositional architectures. TorchDyn further offers step-by-step tutorials and benchmarks designed to guide researchers and contributors.

Michael Poli, Stefano Massaroli, Atsushi Yamashita et al.

The infinite-depth paradigm pioneered by Neural ODEs has launched a renaissance in the search for novel dynamical system-inspired deep learning primitives; however, their utilization in problems of non-trivial size has often proved impossible due to poor computational scalability. This work paves the way for scalable Neural ODEs with time-to-prediction comparable to traditional discrete networks. We introduce hypersolvers, neural networks designed to solve ODEs with low overhead and theoretical guarantees on accuracy. The synergistic combination of hypersolvers and Neural ODEs allows for cheap inference and unlocks a new frontier for practical application of continuous-depth models. Experimental evaluations on standard benchmarks, such as sampling for continuous normalizing flows, reveal consistent pareto efficiency over classical numerical methods.

Stefano Massaroli, Michael Poli, Michelangelo Bin et al.

We introduce a provably stable variant of neural ordinary differential equations (neural ODEs) whose trajectories evolve on an energy functional parametrised by a neural network. Stable neural flows provide an implicit guarantee on asymptotic stability of the depth-flows, leading to robustness against input perturbations and low computational burden for the numerical solver. The learning procedure is cast as an optimal control problem, and an approximate solution is proposed based on adjoint sensivity analysis. We further introduce novel regularizers designed to ease the optimization process and speed up convergence. The proposed model class is evaluated on non-linear classification and function approximation tasks.

Stefano Massaroli, Michael Poli, Jinkyoo Park et al.

Continuous deep learning architectures have recently re-emerged as Neural Ordinary Differential Equations (Neural ODEs). This infinite-depth approach theoretically bridges the gap between deep learning and dynamical systems, offering a novel perspective. However, deciphering the inner working of these models is still an open challenge, as most applications apply them as generic black-box modules. In this work we "open the box", further developing the continuous-depth formulation with the aim of clarifying the influence of several design choices on the underlying dynamics.

Michael Poli, Stefano Massaroli, Junyoung Park et al.

We introduce the framework of continuous--depth graph neural networks (GNNs). Graph neural ordinary differential equations (GDEs) are formalized as the counterpart to GNNs where the input-output relationship is determined by a continuum of GNN layers, blending discrete topological structures and differential equations. The proposed framework is shown to be compatible with various static and autoregressive GNN models. Results prove general effectiveness of GDEs: in static settings they offer computational advantages by incorporating numerical methods in their forward pass; in dynamic settings, on the other hand, they are shown to improve performance by exploiting the geometry of the underlying dynamics.

Michael Poli, Jinkyoo Park, Ilija Ilievski

Finance is a particularly challenging application area for deep learning models due to low noise-to-signal ratio, non-stationarity, and partial observability. Non-deliverable-forwards (NDF), a derivatives contract used in foreign exchange (FX) trading, presents additional difficulty in the form of long-term planning required for an effective selection of start and end date of the contract. In this work, we focus on tackling the problem of NDF tenor selection by leveraging high-dimensional sequential data consisting of spot rates, technical indicators and expert tenor patterns. To this end, we construct a dataset from the Depository Trust & Clearing Corporation (DTCC) NDF data that includes a comprehensive list of NDF volumes and daily spot rates for 64 FX pairs. We introduce WaveATTentionNet (WATTNet), a novel temporal convolution (TCN) model for spatio-temporal modeling of highly multivariate time series, and validate it across NDF markets with varying degrees of dissimilarity between the training and test periods in terms of volatility and general market regimes. The proposed method achieves a significant positive return on investment (ROI) in all NDF markets under analysis, outperforming recurrent and classical baselines by a wide margin. Finally, we propose two orthogonal interpretability approaches to verify noise stability and detect the driving factors of the learned tenor selection strategy.

Stefano Massaroli, Michael Poli, Federico Califano et al.

Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but are solutions of an ordinary differential equation (ODE); however, these networks are still optimized via discrete methods (e.g. gradient descent). In this paper, we explore a different direction: namely, we propose a novel framework for learning in which the parameters themselves are solutions of ODEs. By viewing the optimization process as the evolution of a port-Hamiltonian system, we can ensure convergence to a minimum of the objective function. Numerical experiments have been performed to show the validity and effectiveness of the proposed methods.