Scientific Simulation

Many scientific models are composed of multiple discrete components, and scientists often make heuristic decisions about which components to include. Bayesian inference provides a mathematical framework for systematically selecting model components, but defining prior distributions over model components and developing associated inference schemes has been challenging. We approach this problem in a simulation-based inference framework: We define model priors over candidate components and, from model simulations, train neural networks to infer joint probability distributions over both model components and associated parameters. Our method, simulation-based model inference (SBMI), represents distributions over model components as a conditional mixture of multivariate binary distributions in the Grassmann formalism. SBMI can be applied to any compositional stochastic simulator without requiring likelihood evaluations. We evaluate SBMI on a simple time series model and on two scientific models from neuroscience, and show that it can discover multiple data-consistent model configurations, and that it reveals non-identifiable model components and parameters. SBMI provides a powerful tool for data-driven scientific inquiry which will allow scientists to identify essential model components and make uncertainty-informed modelling decisions.

We present Neural Quantile Estimation (NQE), a novel Simulation-Based Inference (SBI) method based on conditional quantile regression. NQE autoregressively learns individual one dimensional quantiles for each posterior dimension, conditioned on the data and previous posterior dimensions. Posterior samples are obtained by interpolating the predicted quantiles using monotonic cubic Hermite spline, with specific treatment for the tail behavior and multi-modal distributions. We introduce an alternative definition for the Bayesian credible region using the local Cumulative Density Function (CDF), offering substantially faster evaluation than the traditional Highest Posterior Density Region (HPDR). In case of limited simulation budget and/or known model misspecification, a post-processing calibration step can be integrated into NQE to ensure the unbiasedness of the posterior estimation with negligible additional computational cost. We demonstrate that NQE achieves state-of-the-art performance on a variety of benchmark problems.

Computing eigenvalue decomposition (EVD) of a given linear operator, or finding its leading eigenvalues and eigenfunctions, is a fundamental task in many machine learning and scientific simulation problems. For high-dimensional eigenvalue problems, training neural networks to parameterize the eigenfunctions is considered as a promising alternative to the classical numerical linear algebra techniques. This paper proposes a new optimization framework based on the low-rank approximation characterization of a truncated singular value decomposition, accompanied by new techniques called *nesting* for learning the top-$L$ singular values and singular functions in the correct order. The proposed method promotes the desired orthogonality in the learned functions implicitly and efficiently via an unconstrained optimization formulation, which is easy to solve with off-the-shelf gradient-based optimization algorithms. We demonstrate the effectiveness of the proposed optimization framework for use cases in computational physics and machine learning.

This paper presents *Mechanistic Neural Networks*, a neural network design for machine learning applications in the sciences. It incorporates a new *Mechanistic Block* in standard architectures to explicitly learn governing differential equations as representations, revealing the underlying dynamics of data and enhancing interpretability and efficiency in data modeling. Central to our approach is a novel *Relaxed Linear Programming Solver* (NeuRLP) inspired by a technique that reduces solving linear ODEs to solving linear programs. This integrates well with neural networks and surpasses the limitations of traditional ODE solvers enabling scalable GPU parallel processing. Overall, Mechanistic Neural Networks demonstrate their versatility for scientific machine learning applications, adeptly managing tasks from equation discovery to dynamic systems modeling. We prove their comprehensive capabilities in analyzing and interpreting complex scientific data across various applications, showing significant performance against specialized state-of-the-art methods. Source code is available at https://github.com/alpz/mech-nn.

Monotonicity constraints are powerful regularizers in statistical modelling. They can support fairness in computer-aided decision making and increase plausibility in data-driven scientific models. The seminal min-max (MM) neural network architecture ensures monotonicity, but often gets stuck in undesired local optima during training because of partial derivatives being zero when computing extrema. We propose a simple modification of the MM network using strictly-increasing smooth minimum and maximum functions that alleviates this problem. The resulting smooth min-max (SMM) network module inherits the asymptotic approximation properties from the MM architecture. It can be used within larger deep learning systems trained end-to-end. The SMM module is conceptually simple and computationally less demanding than state-of-the-art neural networks for monotonic modelling. Our experiments show that this does not come with a loss in generalization performance compared to alternative neural and non-neural approaches.

Aligning large language models (LLMs) with human values is imperative to mitigate potential adverse effects resulting from their misuse. Drawing from the sociological insight that acknowledging all parties' concerns is a key factor in shaping human values, this paper proposes a novel direction to align LLMs by themselves: social scene simulation. To achieve this, we present MATRIX, a novel social scene simulator that emulates realistic scenes around a user's input query, enabling the LLM to take social consequences into account before responding. MATRIX serves as a virtual rehearsal space, akin to a Monopolylogue, where the LLM performs diverse roles related to the query and practice by itself. To inject this alignment, we fine-tune the LLM with MATRIX-simulated data, ensuring adherence to human values without compromising inference speed. We theoretically show that the LLM with MATRIX outperforms existing methods under mild assumptions. Finally, extensive experiments validate that our method outperforms over 10 baselines across 4 benchmarks. As evidenced by 875 user ratings, our tuned 13B-size LLM exceeds GPT-4 in aligning with human values. See our project page at https://shuotang123.github.io/MATRIX.

Large Language Models have recently gained significant attention in scientific discovery for their extensive knowledge and advanced reasoning capabilities. However, they encounter challenges in effectively simulating observational feedback and grounding it with language to propel advancements in physical scientific discovery. Conversely, human scientists undertake scientific discovery by formulating hypotheses, conducting experiments, and revising theories through observational analysis. Inspired by this, we propose to enhance the knowledge-driven, abstract reasoning abilities of LLMs with the computational strength of simulations. We introduce Scientific Generative Agent (SGA), a bilevel optimization framework: LLMs act as knowledgeable and versatile thinkers, proposing scientific hypotheses and reason about discrete components, such as physics equations or molecule structures; meanwhile, simulations function as experimental platforms, providing observational feedback and optimizing via differentiability for continuous parts, such as physical parameters. We conduct extensive experiments to demonstrate our framework's efficacy in constitutive law discovery and molecular design, unveiling novel solutions that differ from conventional human expectations yet remain coherent upon analysis.

Causal models and methods have great promise, but their progress has been stalled. Proposals using causality get squeezed between two opposing worldviews. Scientific perfectionism--an insistence on only using ``correct'' models--slows the adoption of causal methods in knowledge generating applications. Pushing in the opposite direction, the academic discipline of computer science prefers algorithms with no or few assumptions, and technologies based on automation and scalability are often selected for economic and business applications. We argue that these system-centric inductive biases should be replaced with a human-centric philosophy we refer to as scientific pragmatism. The machine learning community must strike the right balance to make space for the causal revolution to prosper.

Most existing Large Language Model (LLM) benchmarks on scientific problem reasoning focus on problems grounded in high-school subjects and are confined to elementary algebraic operations. To systematically examine the reasoning capabilities required for solving complex scientific problems, we introduce an expansive benchmark suite SciBench for LLMs. SciBench contains a carefully curated dataset featuring a range of collegiate-level scientific problems from mathematics, chemistry, and physics domains. Based on the dataset, we conduct an in-depth benchmarking study of representative open-source and proprietary LLMs with various prompting strategies. The results reveal that current LLMs fall short of delivering satisfactory performance, with the best overall score of merely 43.22%. Furthermore, through a detailed user study, we categorize the errors made by LLMs into ten problem-solving abilities. Our analysis indicates that no single prompting strategy significantly outperforms the others and some strategies that demonstrate improvements in certain problem-solving skills could result in declines in other skills. We envision that SciBench will catalyze further developments in the reasoning abilities of LLMs, thereby ultimately contributing to scientific research and discovery.

The execution of graph algorithms using neural networks has recently attracted significant interest due to promising empirical progress. This motivates further understanding of how neural networks can replicate reasoning steps with relational data. In this work, we study the ability of transformer networks to simulate algorithms on graphs from a theoretical perspective. The architecture we use is a looped transformer with extra attention heads that interact with the graph. We prove by construction that this architecture can simulate individual algorithms such as Dijkstra's shortest path, Breadth- and Depth-First Search, and Kosaraju's strongly connected components, as well as multiple algorithms simultaneously. The number of parameters in the networks does not increase with the input graph size, which implies that the networks can simulate the above algorithms for any graph. Despite this property, we show a limit to simulation in our solution due to finite precision. Finally, we show a Turing Completeness result with constant width when the extra attention heads are utilized.

Computational fluid dynamics (CFD) simulation is an irreplaceable modelling step in many engineering designs, but it is often computationally expensive. Some graph neural network (GNN)-based CFD methods have been proposed. However, the current methods inherit the weakness of traditional numerical simulators, as well as ignore the cell characteristics in the mesh used in the finite volume method, a common method in practical CFD applications. Specifically, the input nodes in these GNN methods have very limited information about any object immersed in the simulation domain and its surrounding environment. Also, the cell characteristics of the mesh such as cell volume, face surface area, and face centroid are not included in the message-passing operations in the GNN methods. To address these weaknesses, this work proposes two novel geometric representations: Shortest Vector (SV) and Directional Integrated Distance (DID). Extracted from the mesh, the SV and DID provide global geometry perspective to each input node, thus removing the need to collect this information through message-passing. This work also introduces the use of Finite Volume Features (FVF) in the graph convolutions as node and edge attributes, enabling its message-passing operations to adjust to different nodes. Finally, this work is the first to demonstrate how residual training, with the availability of low-resolution data, can be adopted to improve the flow field prediction accuracy. Experimental results on two datasets with five different state-of-the-art GNN methods for CFD indicate that SV, DID, FVF and residual training can effectively reduce the predictive error of current GNN-based methods by as much as 41%. Our codes and datasets are available at https://github.com/toggled/FvFGeo.

Amortized variational inference is an often employed framework in simulation-based inference that produces a posterior approximation that can be rapidly computed given any new observation. Unfortunately, there are few guarantees about the quality of these approximate posteriors. We propose Conformalized Amortized Neural Variational Inference (CANVI), a procedure that is scalable, easily implemented, and provides guaranteed marginal coverage. Given a collection of candidate amortized posterior approximators, CANVI constructs conformalized predictors based on each candidate, compares the predictors using a metric known as predictive efficiency, and returns the most efficient predictor. CANVI ensures that the resulting predictor constructs regions that contain the truth with a user-specified level of probability. CANVI is agnostic to design decisions in formulating the candidate approximators and only requires access to samples from the forward model, permitting its use in likelihood-free settings. We prove lower bounds on the predictive efficiency of the regions produced by CANVI and explore how the quality of a posterior approximation relates to the predictive efficiency of prediction regions based on that approximation. Finally, we demonstrate the accurate calibration and high predictive efficiency of CANVI on a suite of simulation-based inference benchmark tasks and an important scientific task: analyzing galaxy emission spectra.

Bootstrap is a popular methodology for simulating input uncertainty. However, it can be computationally expensive when the number of samples is large. We propose a new approach called **Orthogonal Bootstrap** that reduces the number of required Monte Carlo replications. We decomposes the target being simulated into two parts: the *non-orthogonal part* which has a closed-form result known as Infinitesimal Jackknife and the *orthogonal part* which is easier to be simulated. We theoretically and numerically show that Orthogonal Bootstrap significantly reduces the computational cost of Bootstrap while improving empirical accuracy and maintaining the same width of the constructed interval.

Amortized Bayesian inference trains neural networks to solve stochastic inference problems using model simulations, thereby making it possible to rapidly perform Bayesian inference for any newly observed data. However, current simulation-based amortized inference methods are simulation-hungry and inflexible: They require the specification of a fixed parametric prior, simulator, and inference tasks ahead of time. Here, we present a new amortized inference method---the Simformer---which overcomes these limitations. By training a probabilistic diffusion model with transformer architectures, the Simformer outperforms current state-of-the-art amortized inference approaches on benchmark tasks and is substantially more flexible: It can be applied to models with function-valued parameters, it can handle inference scenarios with missing or unstructured data, and it can sample arbitrary conditionals of the joint distribution of parameters and data, including both posterior and likelihood. We showcase the performance and flexibility of the Simformer on simulators from ecology, epidemiology, and neuroscience, and demonstrate that it opens up new possibilities and application domains for amortized Bayesian inference on simulation-based models.

Publications proposing novel machine learning methods are often primarily rated by exhibited predictive performance on selected problems. In this position paper we argue that predictive performance alone is not a good indicator for the worth of a publication. Using it as such even fosters problems like inefficiencies of the machine learning research community as a whole and setting wrong incentives for researchers. We therefore put out a call for the publication of ``negative'' results, which can help alleviate some of these problems and improve the scientific output of the machine learning research community. To substantiate our position, we present the advantages of publishing negative results and provide concrete measures for the community to move towards a paradigm where their publication is normalized.