Mathematical concepts emerge through an interplay of processes, including experimentation, efforts at proof, and counterexamples. In this paper, we present a new multi-agent model for computational mathematical discovery based on this observation. Our system, conceived with research in mind, poses its own conjectures and then attempts to prove them, making decisions informed by this feedback and an evolving data distribution. Inspired by the history of Euler's conjecture for polyhedra and an open challenge in the literature, we benchmark with the task of autonomously recovering the concept of homology from polyhedral data and knowledge of linear algebra. Our system completes this learning problem. Most importantly, the experiments are ablations, statistically testing the value of the complete dynamic and controlling for experimental setup. They support our main claim: that the optimisation of the right combination of local processes can lead to surprisingly well-aligned notions of mathematical interestingness.

Many cancer monotherapies demonstrate limited clinical efficacy, making combination therapies a relevant treatment strategy. The extensive number of potential drug combinations and context-specific response profiles complicates the prediction of drug combination responses. Existing computational models are typically trained to predict a single aggregated synergy score, which summarises drug responses across different dosage combinations, such as Bliss or Loewe scores. This oversimplification of the drug-response surface leads to high prediction uncertainty and limited actionability, as these models fail to distinguish between potency and efficacy. We introduce DeepSynBa, an actionable model that predicts the complete dose-response matrix of drug pairs instead of relying on an aggregated synergy score. This is achieved by predicting parameters describing the response surface as an intermediate layer in the model. Evaluated on the NCI-ALMANAC and the O'Neil datasets, DeepSynBa outperforms the state-of-the-art methods in the dose-response matrix prediction task across most evaluation scenarios, including testing on novel drug combinations, cell lines, and drugs, across nine different tissue types. We also show that DeepSynBa yields reliable synergy score predictions. More importantly, DeepSynBa can predict drug combination responses across different dosages for untested combinations. The intermediate dose-response parameter layer enables the separation of efficacy from potency, informing the selection of dosage ranges that optimise efficacy while limiting off-target toxicity in experimental screens. The predictive capability and the downstream actionability make DeepSynBa a powerful tool for advancing drug combination research beyond the limitations of the current approaches. The code and the dataset for DeepSynBa are available at https://github.com/hikuru/DeepSynBa.

Modern generative AI models, such as diffusion and flow matching models, can sample from rich data distributions. However, many applications, especially in science and engineering, require more than drawing samples from the model distribution: they require searching within this distribution for samples that optimise task-specific criteria. In this work, we propose O3 (Optimisation Over the Outputs of Generative Models), a method for sample-efficient black-box optimisation over continuous-variable diffusion and flow-matching models. O3 is built around surrogate latent spaces: low-dimensional Euclidean embeddings that can be extracted from a generative model without additional training. The resulting representations have controllable dimensionality and support the direct application of standard optimisation algorithms. We show, on image and protein design tasks, that surrogate-space optimisation finds substantially higher-scoring samples than standard sampling or optimisation in the original latent space. Our method is model- and optimiser-agnostic, incurs negligible additional cost over standard generation, and requires no retraining or fine-tuning of the generative model.

Variational mean field approximations tend to struggle with contemporary overparametrized deep neural networks. Where a Bayesian treatment is usually associated with high-quality predictions and uncertainties, the practical reality has been the opposite, with unstable training, poor predictive power, and subpar calibration. Building upon recent work on reparametrizations of neural networks, we propose a simple variational family that considers two independent linear subspaces of the parameter space. These represent functional changes inside and outside the support of training data. This allows us to build a fully-correlated approximate posterior reflecting the overparametrization that tunes easy-to-interpret hyperparameters. We develop scalable numerical routines that maximize the associated evidence lower bound (ELBO) and sample from the approximate posterior. Empirically, we observe state-of-the-art performance across tasks, models, and datasets compared to a wide array of baseline methods. Our results show that approximate Bayesian inference applied to deep neural networks is far from a lost cause when constructing inference mechanisms that reflect the geometry of reparametrizations.

We present a method that models the evolution of an unbounded number of time series clusters by switching among an unknown number of regimes with linear dynamics. We develop a Bayesian non-parametric approach using a hierarchical Dirichlet process as a prior on the parameters of a Switching Linear Dynamical System and a Gaussian process prior to model the statistical variations in amplitude and temporal alignment within each cluster. By modeling the evolution of time series patterns, the method avoids unnecessary proliferation of clusters in a principled manner. We perform inference by formulating a variational lower bound for off-line and on-line scenarios, enabling efficient learning through optimization. We illustrate the versatility and effectiveness of the approach through several case studies of electrocardiogram analysis using publicly available databases.

Reinforcement learning is a general method for learning in sequential settings, but it can often be difficult to specify a good reward function when the task is complex. In these cases, preference feedback or expert demonstrations can be used instead. However, existing approaches utilising both together are often ad-hoc, rely on domain-specific properties, or won't scale. We develop a new framing for learning from human data, reward-rational partial orderings over observations, designed to be flexible and scalable. Based on this we introduce a practical algorithm, LEOPARD: Learning Estimated Objectives from Preferences And Ranked Demonstrations. LEOPARD can learn from a broad range of data, including negative demonstrations, to efficiently learn reward functions across a wide range of domains. We find that when a limited amount of preference and demonstration feedback is available, LEOPARD outperforms existing baselines by a significant margin. Furthermore, we use LEOPARD to investigate learning from many types of feedback compared to just a single one, and find that combining feedback types is often beneficial.

Sampling from generative models has become a crucial tool for applications like data synthesis and augmentation. Diffusion, Flow Matching and Continuous Normalising Flows have shown effectiveness across various modalities, and rely on latent variables for generation. For experimental design or creative applications that require more control over the generation process, it has become common to manipulate the latent variable directly. However, existing approaches for performing such manipulations (e.g. interpolation or forming low-dimensional representations) only work well in special cases or are network or data-modality specific. We propose Latent Optimal Linear combinations (LOL) as a general-purpose method to form linear combinations of latent variables that adhere to the assumptions of the generative model. As LOL is easy to implement and naturally addresses the broader task of forming any linear combinations, e.g. the construction of subspaces of the latent space, LOL dramatically simplifies the creation of expressive low-dimensional representations of high-dimensional objects.

The growing integration of large language models across professional domains transforms how experts make critical decisions in healthcare, education, and law. While significant research effort focuses on getting these systems to communicate their outputs with probabilistic measures of reliability, many consequential forms of uncertainty in professional contexts resist such quantification. A physician pondering the appropriateness of documenting possible domestic abuse, a teacher assessing cultural sensitivity, or a mathematician distinguishing procedural from conceptual understanding face forms of uncertainty that cannot be reduced to percentages. This paper argues for moving beyond simple quantification toward richer expressions of uncertainty essential for beneficial AI integration. We propose participatory refinement processes through which professional communities collectively shape how different forms of uncertainty are communicated. Our approach acknowledges that uncertainty expression is a form of professional sense-making that requires collective development rather than algorithmic optimization.

Aims:Machine learning (ML) algorithms applied to the electrocardiography (ECG) have been successful in several cardiac diagnoses, however, rarely been used for the diagnostics of takotsubo syndrome (TTS). Our aim was to develop ML-based ECG-models to differentiate TTS from patients with myocardial infarction (MI). Methods and results:Cross-sectional study in Stockholm. A neural network with UNet architecture was trained and validated on 507 TTS cases and 14 978 controls with suspected and verified MI, identified from the Swedish coronary angiography and angioplasty register. Cross-validation was performed. The models were compared with cardiologists using previously proposed ECG criteria. Receiver operating characteristics (ROC) area under the curve (AUC) for discriminating TTS from patients with ST-elevation and non-ST-elevation MI ROC AUC 0.88 (cross-validation: 0.85-0.92) and 0.86 (cross-validation: 0.82-0.91), respectively. ROC AUC for discriminating TTS from verified MI [non-ST-elevation MI (NSTEMI) and ST-elevation MI (STEMI)] was 0.87 (cross-validation: 0.83-0.91) with sensitivity (0.75) and specificity (0.83) with low positive predictive value (PPV) and high negative predictive value (NPV). Results for suspected MI was ROC AUC 0.85 (cross validation: 0.81-0.91) with sensitivity (0.75) and specificity (0.79) with low PPV (0.11) and high NPV (0.99). The committee of two cardiologists using a combination of ECG criteria achieved an ROC AUC of 0.71. Conclusion:Machine learning models could discriminate TTS from MI (NSTEMI and STEMI) and suspected MI with high sensitivity and NPV, outperforming cardiologists using conventional criteria. The models require further refinement to increase PPV, precision-recall and external validation, but it holds promise for TTS screening aiding the clinician in ruling out TTS.

Thompson sampling (TS) is a powerful and widely used strategy for sequential decision-making, with applications ranging from Bayesian optimization to reinforcement learning (RL). Despite its success, the theoretical foundations of TS remain limited, particularly in settings with complex temporal structure such as RL. We address this gap by establishing no-regret guarantees for TS using models with Gaussian marginal distributions. Specifically, we consider TS in episodic RL with joint Gaussian process (GP) priors over rewards and transitions. We prove a regret bound of $\mathcal{\tilde{O}}(\sqrt{KH\Gamma(KH)})$ over $K$ episodes of horizon $H$, where $\Gamma(\cdot)$ captures the complexity of the GP model. Our analysis addresses several challenges, including the non-Gaussian nature of value functions and the recursive structure of Bellman updates, and extends classical tools such as the elliptical potential lemma to multi-output settings. This work advances the understanding of TS in RL and highlights how structural assumptions and model uncertainty shape its performance in finite-horizon Markov Decision Processes.

Many cancer monotherapies have limited activity in clinic, making combinations a relevant treatment strategy. The number of possible combinations is vast, and the responses can be context-specific, making it challenging to predict combination effects. Existing computational models typically predict a single aggregated synergy score, i.e. Bliss or Loewe, for a given drug combination. However, these approaches exhibit high prediction uncertainty and limited actionability because they fail to differentiate between potency and efficacy by oversimplifying the drug-response surface using a single synergy score. To address these limitations, we introduce DeepSynBa, which models the full dose-response matrix of drug pairs rather than an aggregated synergy score. DeepSynBa formulates this task as a regression problem and uses cell line-specific features and drug embeddings to predict the entire drug-response matrix within a deep learning framework. Following SynBa’s approach of modelling the dose-response surface [1], DeepSynBa includes an intermediate layer that estimates pharmacological parameters, which are then used to predict dose-response values across dosages. This design also enables post-hoc calculation of traditional synergy scores like Loewe and Bliss, maintaining compatibility with existing synergy predictors. We evaluated DeepSynBa on the NCI-ALMANAC dataset, which contains 104 FDA-approved cancer drugs, 60 cell lines, and nearly 300, 000 drug pair-cell line combinations. DeepSynBa outperforms comboFM [2] and comboLTR [3], two state-of-the-art methods for direct dose-response matrix prediction across multiple evaluation scenarios, such as novel drug combinations, cell lines, and drugs. DeepSynBa achieves RMSE values of 7.72 and 7.71 in the new drug combination and new cell line settings, respectively, which is an improvement of approximately 65% compared to comboFM, the next-best model. DeepSynBa also consistently attained lower RMSE and higher Pearson and Spearman correlations across nine tissue types including breast, colon, and leukemia. DeepSynBa provides a more accurate framework for predicting the full dose-response matrix than existing methods. Additionally, DeepSynBa supports experimental design by providing detailed predictions of drug combination responses across dosages. For untested combinations, by utilizing the parameter layer which separates efficacy from potency, biologists can select dosage ranges that optimize efficacy while limiting off-target toxicity, hence increasing the likelihood of identifying optimal therapeutic dosages. 1.Zhang H. et al. Bioinformatics, 39, 1:i121-i130 (2023).2.Julkunen H. et al. Nature Communications, 11, 1:1-11 (2020).3.Wang T. et al. Bioinformatics, 37, 1:i93-i101(2021). Marta Milo, Haoting Zhang, Halil ibrahim Kuru, A. Ercument Cicek, Oznur Tastan, Carl Henrik Ek, Magnus Rattray. DeepSynBa: A deep learning method to improve drug combination predictions using full dose-response matrices and contextual features of cell lines and cancer indications [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2025; Part 1 (Regular Abstracts); 2025 Apr 25-30; Chicago, IL. Philadelphia (PA): AACR; Cancer Res 2025;85(8_Suppl_1):Abstract nr 3646.

Riemannian geometry provides us with powerful tools to explore the latent space of generative models while preserving the underlying structure of the data. The latent space can be equipped it with a Riemannian metric, pulled back from the data manifold. With this metric, we can systematically navigate the space relying on geodesics defined as the shortest curves between two points. Generative models are often stochastic, causing the data space, the Riemannian metric, and the geodesics, to be stochastic as well. Stochastic objects are at best impractical, and at worst impossible, to manipulate. A common solution is to approximate the stochastic pullback metric by its expectation. But the geodesics derived from this expected Riemannian metric do not correspond to the expected length-minimising curves. In this work, we propose another metric whose geodesics explicitly minimise the expected length of the pullback metric. We show this metric defines a Finsler metric, and we compare it with the expected Riemannian metric. In high dimensions, we prove that both metrics converge to each other at a rate of $O\left(\frac{1}{D}\right)$. This convergence implies that the established expected Riemannian metric is an accurate approximation of the theoretically more grounded Finsler metric. This provides justification for using the expected Riemannian metric for practical implementations.

In recent years, the importance of digitalization techniques in the field of engineering has grown rapidly and led to many new applications, allowing analyzing, predicting, and reusing data. An essential type of engineering data is related to finite element models frequently used for simulation, where many small geometric changes occur during a product’s development process. These modifications reflect the engineer’s implicit knowledge, as they show the engineer’s decisions during the development process. While numerous approaches exist to formalize geometric information as a whole, representing only the changes between finite element models, and thus engineers’ knowledge, has not been investigated yet. In this work, a representation of geometric engineering changes between finite element models is presented, which is automatically learned from past engineering data. The representation is based on vectors between variants, whose dimensionality is reduced with a novel directed point cloud autoencoder. It is shown that this representation of change preserves similarities and differences between different geometric changes while being suitable as input for further machine learning applications, enabling further use of the acquired knowledge. To summarize, the objective is to learn from changes and their motivations in finished engineering developments and to transfer the insights to future design processes, i.e., to inspire engineers and make processes more efficient and goal-oriented.

Much of an engineer's implicit knowledge and experience is embedded in data from previous engineering processes. In the typical automotive structural design process, many simulation variants are created to incrementally improve relevant performance values. Engineers decide about the modifications to make to the geometry to achieve these improvements. The steps of this development process reflect the engineer's decisions and thus her/his knowledge and experience. This implicit knowledge is potentially helpful for new projects, but is not easily accessible. This paper proposes a method to automatically extract past solutions from a database containing finite element simulation data from previous development processes. A reference model with high similarity to the problem at hand is identified in the database. Since the development process of this model is already completed, there must exist a later version of the model where the problem has been solved. The difference between the reference model and the identified improved model represents an engineer's knowledge that was used to solve a similar problem in the past. Possible solutions identified this way are successfully used to support engineers in their decisions in new projects during the structural design process. The method is applied to pedestrian safety data from the structural design process of cars using real-world data.

Current approximate posteriors in Bayesian neural networks (BNNs) exhibit a crucial limitation: they fail to maintain invariance under reparameterization, i.e. BNNs assign different posterior densities to different parametrizations of identical functions. This creates a fundamental flaw in the application of Bayesian principles as it breaks the correspondence between uncertainty over the parameters with uncertainty over the parametrized function. In this paper, we investigate this issue in the context of the increasingly popular linearized Laplace approximation. Specifically, it has been observed that linearized predictives alleviate the common underfitting problems of the Laplace approximation. We develop a new geometric view of reparametrizations from which we explain the success of linearization. Moreover, we demonstrate that these reparameterization invariance properties can be extended to the original neural network predictive using a Riemannian diffusion process giving a straightforward algorithm for approximate posterior sampling, which empirically improves posterior fit.

The close relation between spatial kinematics and line geometry has been proven to be fruitful in surface detection and reconstruction. However, methods based on this approach are limited to simple geometric shapes that can be formulated as a linear subspace of line or line element space. The core of this approach is a principal component formulation to find a best-fit approximant to a possibly noisy or impartial surface given as an unordered set of points or point cloud. We expand on this by introducing the Gaussian process latent variable model, a probabilistic non-linear non-parametric dimensionality reduction approach following the Bayesian paradigm. This allows us to find structure in a lower dimensional latent space for the surfaces of interest. We show how this can be applied in surface approximation and unsupervised segmentation to the surfaces mentioned above and demonstrate its benefits on surfaces that deviate from these. Experiments are conducted on synthetic and real-world objects.

Motivation: There exists a range of different quantification frameworks to estimate the synergistic effect of drug combinations. The diversity and disagreement in estimates make it challenging to determine which combinations from a large drug screening should be proceeded with. Furthermore, the lack of accurate uncertainty quantification for those estimates precludes the choice of optimal drug combinations based on the most favourable synergistic effect.Results: In this work, we propose SynBa, a flexible Bayesian approach to estimate the uncertainty of the synergistic efficacy and potency of drug combinations, so that actionable decisions can be derived from the model outputs. The actionability is enabled by incorporating the Hill equation into SynBa, so that the parameters representing the potency and the efficacy can be preserved. Existing knowledge may be conveniently inserted due to the flexibility of the prior, as shown by the empirical Beta prior defined for the normalized maximal inhibition. Through experiments on large combination screenings and comparison against benchmark methods, we show that SynBa provides improved accuracy of dose-response predictions and better-calibrated uncertainty estimation for the parameters and the predictions.Availability and implementationThe code for SynBa is available at . The datasets are publicly available (DOI of DREAM: ; DOI of the NCI-ALMANAC subset: 1).

Model-based reinforcement learning (RL) algorithms do not typically consider environments with multiple dynamic modes, where it is beneficial to avoid inoperable or undesirable modes. We present a model-based RL algorithm that constrains training to a single dynamic mode with high probability. This is a difficult problem because the mode constraint is a hidden variable associated with the environment's dynamics. As such, it is 1) unknown a priori and 2) we do not observe its output from the environment, so cannot learn it with supervised learning. We present a nonparametric dynamic model which learns the mode constraint alongside the dynamic modes. Importantly, it learns latent structure that our planning scheme leverages to 1) enforce the mode constraint with high probability, and 2) escape local optima induced by the mode constraint. We validate our method by showing that it can solve a simulated quadcopter navigation task whilst providing a level of constraint satisfaction both during and after training.