The development of efficient machine learning models for molecular systems representation is becoming crucial in scientific research. We introduce Tensor-Net, an innovative O(3)-equivariant message-passing neural network architecture that leverages Cartesian tensor representations. By using Cartesian tensor atomic embeddings, feature mixing is simplified through matrix product operations. Furthermore, the cost-effective decomposition of these tensors into rotation group irreducible representations allows for the separate processing of scalars, vectors, and tensors when necessary. Compared to higher-rank spherical tensor models, TensorNet demonstrates state-of-the-art performance with significantly fewer parameters. For small molecule potential energies, this can be achieved even with a single interaction layer. As a result of all these properties, the model's computational cost is substantially decreased. Moreover, the accurate prediction of vector and tensor molecular quantities on top of potential energies and forces is possible. In summary, TensorNet's framework opens up a new space for the design of state-of-the-art equivariant models.

Conventional statistical wisdom established a well-understood relationship between model complexity and prediction error, typically presented as a U-shaped curve reflecting a transition between under-and overfitting regimes. However, motivated by the success of overparametrized neural networks, recent influential work has suggested this theory to be generally incomplete, introducing an additional regime that exhibits a second descent in test error as the parameter count p grows past sample size n - a phenomenon dubbed double descent. While most attention has naturally been given to the deep-learning setting, double descent was shown to emerge more generally across non-neural models: known cases include linear regression, trees, and boosting. In this work, we take a closer look at the evidence surrounding these more classical statistical machine learning methods and challenge the claim that observed cases of double descent truly extend the limits of a traditional U-shaped complexity-generalization curve therein. We show that once careful consideration is given to what is being plotted on the x-axes of their double descent plots, it becomes apparent that there are implicitly multiple, distinct complexity axes along which the parameter count grows. We demonstrate that the second descent appears exactly (and only) when and where the transition between these underlying axes occurs, and that its location is thus not inherently tied to the interpolation threshold p=n. We then gain further insight by adopting a classical nonparametric statistics perspective. We interpret the investigated methods as smoothers and propose a generalized measure for the effective number of parameters they use on unseen examples, using which we find that their apparent double descent curves do indeed fold back into more traditional convex shapes - providing a resolution to the ostensible tension between double descent and traditional statistical intuition.

We study the problems of estimating the past and future evolutions of two diffusion processes that spread concurrently on a network. Specifically, given a known network $G=(V, \overrightarrow{E})$ and a (possibly noisy) snapshot $\mathcal{O}_n$ of its state taken at (a possibly unknown) time $W$, we wish to determine the posterior distributions of the initial state of the network and the infection times of its nodes. These distributions are useful in finding source nodes of epidemics and rumors -- $\textit{backward inference}$ -- , and estimating the spread of a fixed set of source nodes -- $\textit{forward inference}$. To model the interaction between the two processes, we study an extension of the independent-cascade (IC) model where, when a node gets infected with either process, its susceptibility to the other one changes. First, we derive the exact joint probability of the initial state of the network and the observation-snapshot $\mathcal{O}_n$. Then, using the machinery of factor-graphs, factor-graph transformations, and the generalized distributive-law, we derive a Belief-Propagation (BP) based algorithm that is scalable to large networks and can converge on graphs of arbitrary topology (at a likely expense in approximation accuracy).

We investigate the challenge of task planning for multi-task embodied agents in open-world environments. Two main difficulties are identified: 1) executing plans in an open-world environment (e.g., Minecraft) necessitates accurate and multi-step reasoning due to the long-term nature of tasks, and 2) as vanilla planners do not consider how easy the current agent can achieve a given sub-task when ordering parallel sub-goals within a complicated plan, the resulting plan could be inefficient or even infeasible. To this end, we propose "$\underline{D}$escribe, $\underline{E}$xplain, $\underline{P}$lan and $\underline{S}$elect" ($\textbf{DEPS}$), an interactive planning approach based on Large Language Models (LLMs). DEPS facilitates better error correction on initial LLM-generated $\textit{plan}$ by integrating $\textit{description}$ of the plan execution process and providing self-$\textit{explanation}$ of feedback when encountering failures during the extended planning phases. Furthermore, it includes a goal $\textit{selector}$, which is a trainable module that ranks parallel candidate sub-goals based on the estimated steps of completion, consequently refining the initial plan. Our experiments mark the milestone of the first zero-shot multi-task agent that can robustly accomplish 70+ Minecraft tasks and nearly double the overall performances. Further testing reveals our method's general effectiveness in popularly adopted non-open-ended domains as well (i.e., ALFWorld and tabletop manipulation). The ablation and exploratory studies detail how our design beats the counterparts and provide a promising update on the $\texttt{ObtainDiamond}$ grand challenge with our approach. The code is released at https://github.com/CraftJarvis/MC-Planner.

We propose a novel approach for domain generalisation (DG) leveraging risk distributions to characterise domains, thereby achieving domain invariance. In our findings, risk distributions effectively highlight differences between training domains and reveal their inherent complexities. In testing, we may observe similar, or potentially intensifying in magnitude, divergences between risk distributions. Hence, we propose a compelling proposition: Minimising the divergences between risk distributions across training domains leads to robust invariance for DG. The key rationale behind this concept is that a model, trained on domain-invariant or stable features, may consistently produce similar risk distributions across various domains. Building upon this idea, we propose Risk Distribution Matching (RDM). Using the maximum mean discrepancy (MMD) distance, RDM aims to minimise the variance of risk distributions across training domains. However, when the number of domains increases, the direct optimisation of variance leads to linear growth in MMD computations, resulting in inefficiency. Instead, we propose an approximation that requires only one MMD computation, by aligning just two distributions: that of the worst-case domain and the aggregated distribution from all domains. Notably, this method empirically outperforms optimising distributional variance while being computationally more efficient. Unlike conventional DG matching algorithms, RDM stands out for its enhanced efficacy by concentrating on scalar risk distributions, sidestepping the pitfalls of high-dimensional challenges seen in feature or gradient matching. Our extensive experiments on standard benchmark datasets demonstrate that RDM shows superior generalisation capability over state-of-the-art DG methods.

Normalizing flow is a class of deep generative models for efficient sampling and likelihood estimation, which achieves attractive performance, particularly in high dimensions. The flow is often implemented using a sequence of invertible residual blocks. Existing works adopt special network architectures and regularization of flow trajectories. In this paper, we develop a neural ODE flow network called JKO-iFlow, inspired by the Jordan-Kinderleherer-Otto (JKO) scheme, which unfolds the discrete-time dynamic of the Wasserstein gradient flow. The proposed method stacks residual blocks one after another, allowing efficient block-wise training of the residual blocks, avoiding sampling SDE trajectories and score matching or variational learning, thus reducing the memory load and difficulty in end-to-end training. We also develop adaptive time reparameterization of the flow network with a progressive refinement of the induced trajectory in probability space to improve the model accuracy further. Experiments with synthetic and real data show that the proposed JKO-iFlow network achieves competitive performance compared with existing flow and diffusion models at a significantly reduced computational and memory cost.

Fourier neural operators (FNOs) can learn highly nonlinear mappings between function spaces, and have recently become a popular tool for learning responses of complex physical systems. However, to achieve good accuracy and efficiency, FNOs rely on the Fast Fourier transform (FFT), which is restricted to modeling problems on rectangular domains. To lift such a restriction and permit FFT on irregular geometries as well as topology changes, we introduce domain agnostic Fourier neural operator (DAFNO), a novel neural operator architecture for learning surrogates with irregular geometries and evolving domains. The key idea is to incorporate a smoothed characteristic function in the integral layer architecture of FNOs, and leverage FFT to achieve rapid computations, in such a way that the geometric information is explicitly encoded in the architecture. In our empirical evaluation, DAFNO has achieved state-of-the-art accuracy as compared to baseline neural operator models on two benchmark datasets of material modeling and airfoil simulation. To further demonstrate the capability and generalizability of DAFNO in handling complex domains with topology changes, we consider a brittle material fracture evolution problem. With only one training crack simulation sample, DAFNO has achieved generalizability to unseen loading scenarios and substantially different crack patterns from the trained scenario.

The rapidly advancing fields of statistical modeling and machine learning have significantly enhanced data-driven design and optimization. This paper focuses on leveraging these design algorithms to optimize a medical walker, an integral part of gait rehabilitation and physiological therapy of the lower extremities. To achieve the desirable qualities of a walker, we train a predictive machine-learning model to identify trade-offs between performance objectives, thus enabling the use of efficient optimization algorithms. To do this, we use an Automated Machine Learning model utilizing a stacked-ensemble approach shown to outperform traditional ML models. However, training a predictive model requires vast amounts of data for accuracy. Due to limited publicly available walker designs, this paper presents a dataset of more than 5,000 parametric walker designs with performance values to assess mass, structural integrity, and stability. These performance values include displacement vectors for the given load case, stress coefficients, mass, and other physical properties. We also introduce a novel method of systematically calculating the stability index of a walker. We use MultiObjective Counterfactuals for Design (MCD), a novel genetic-based optimization algorithm, to explore the diverse 16-dimensional design space and search for high-performing designs based on numerous objectives. This paper presents potential walker designs that demonstrate up to a 30% mass reduction while increasing structural stability and integrity. This work takes a step toward the improved development of assistive mobility devices.

The Kohn-Sham equations underlie many important applications such as the discovery of new catalysts. Recent machine learning work on catalyst modeling has focused on prediction of the energy, but has so far not yet demonstrated significant out-of-distribution generalization. Here we investigate another approach based on the pointwise learning of the Kohn-Sham charge-density. On a new dataset of bulk catalysts with charge densities, we show density models can generalize to new structures with combinations of elements not seen at train time, a form of combinatorial generalization. We show that over 80% of binary and ternary test cases achieve faster convergence than standard baselines in Density Functional Theory, amounting to an average reduction of 13% in the number of iterations required to reach convergence, which may be of independent interest. Our results suggest that density learning is a viable alternative, trading greater inference costs for a step towards combinatorial generalization, a key property for applications.

There is a current increase in the development of "4D" Doppler-capable radar and lidar range sensors that produce 3D point clouds where all points also have information about the radial velocity relative to the sensor. 4D radars in particular are interesting for object perception and navigation in low-visibility conditions (dust, smoke) where lidars and cameras typically fail. With the advent of high-resolution Doppler-capable radars comes the possibility of estimating odometry from single point clouds, foregoing the need for scan registration which is error-prone in feature-sparse field environments. We compare several odometry estimation methods, from direct integration of Doppler/IMU data and Kalman filter sensor fusion to 3D scan-to-scan and scan-to-map registration, on three datasets with data from two recent 4D radars and two IMUs. Surprisingly, our results show that the odometry from Doppler and IMU data alone give similar or better results than 3D point cloud registration. In our experiments, the average position error can be as low as 0.3% over 1.8 and 4.5km trajectories. That allows accurate estimation of 6DOF ego-motion over long distances also in feature-sparse mine environments. These results are useful not least for applications of navigation with resource-constrained robot platforms in feature-sparse and low-visibility conditions such as mining, construction, and search & rescue operations.