sylph is ostensibly a system for biological analysis, but its core is a system for automated geometric proofs about dynamical systems. i'm going to try to demystify that a bit in this post, and link it to the literature i'm drawing all this from. i'm still prototyping this, so it's not quite solid enough to just "write out the math", but thankfully the preceding work has been very good at describing the perspectives. at least, that's my excuse for dodging pedagogy in this post.
we call that backend proof system and numerical engine "paracelsus". it was developed from some past work i did in stable integration and targeting schemes for low-energy lunar transfers. about a year ago, vivien encouraged me to look at the applied math aspects of systems biology, and i was very amused to discover how much overlap there was, right down to the relationship between phase space structures and model optimization.
this post briefly explains the direction i've taken, what i haven't done, and how it compares to other modes of analysis. as this work solidifies, i'll be making more posts here about specific methods in use, what parts didn't work out, and how this system integrates in common biology workflows.
Quantitative modeling is poised to redefine the practice of biology. Biological systems are messy and interconnected, filled with feedback loops and indirect action, making it computationally infeasible to model a large system with high fidelity. One strategy to handle this is to carve out small pieces of the system and hope that studying them in exacting detail devoid of context will produce results close enough to the real thing. Another is to treat the system as a black box, collections of inputs and outputs to run statistical regressions over, trying to establish causal relationships with no understanding of the mechanisms inside.
However, there is a third approach: to model mathematically the important parts of the system and their relationships, capturing as much essential complexity as possible while abstracting over aspects that aren't critical. These models are very useful, but biological systems are intricate and highly connected, so they can quickly become intractable as more functional relationships are determined to be relevant. However, that's only because the mathematics underlying these biological simulations is overly cautious and dated. Using modern mathematical perspective from proof theory and computational differential geometry, these equations may be rearranged into equivalent simpler forms, sliced up into pieces and run in parallel, and in some cases contracted down to trivial logical problems. These techniques have already seen common use in astrodynamics, computational fluid mechanics, and computational physics. We're taking the tools that have enabled rapid growth and development in these fields and applying them to biological analysis.